- Email: [email protected]
Effects of time dependence of multiplication process on avalanche noise
EFFECTS OF TIME DEPENDENCE OF MULTIPLICATION PROCESS ON AVALANCHE NOISE* I. M. NAQVI Department
of Electrical (Received
Engineering,
University
of Hawaii,
13 April 1972; in recisedform
Honolulu,
Hawaii,
U.S.A.
12 June 1972)
Abstract-Theoretical and experimental studies of noise generated due to the randomness of the multiplication process in the avalanche region of a uniform diode are presented. The theory extends the results of McIntyre to include the time dependence of the multiplication process. It also shows the correspondence between the results of MCIntyre, Gummel and Blue, Hines and Tager. The spacecharge feedback and transit-time effects have been neglected in this analysis. The theoretical and the experimental results described have shown that even at frequencies well below transit-time frequency, the importance of the factor resulting from consideration of the time dependence of the multiplication process cannot be ignored. The measurements of the avalanche noise on uniform p+--n--n+ silicon diodes are found to be in good agreement with the theory presented here.
1. BACKGROUND
the mechanism for noise generation. In his analysis, McIntyre has assumed that under the lowfrequency limits (frequencies well below transittime frequency), only the d.c. multiplication factor M need be considered. The theory presented in this paper shows that this assumption is justifiable only at low multiplications when o*M*? -=s 1, where o is the frequency, and 7 is the intrinsic response time of carriers. The experimental results showing the agreement with McIntyre’s theory at low multiplication have been reported by several authors [ lo- 121. Hines’[3] expression for noise in the avalanche region is derived by qualitative reasoning. It contains the very simplified assumptions of equal ionization rates and velocities of holes and electrons. The application of this expression to silicon diodes, therefore, suffers from these approximations. The expression for noise derived by Tager [4] also considers the assumptions of equal ionization rates and velocities of electrons and holes. His expression, however, does include the time dependence of the multiplication process. The intrinsic response time obtained by Tager under the above crssumptions is different from those derived in this paper only by the factor arising from the carrier induced displacement current.
describes new theoretical and experimental studies of noise generated due to the multiplication process in the avalanche region of uniform diodes. It extends the theoretical results of McIntyre[ l] by including the time dependence of the multiplication process in the avalanche region. It also shows the correspondence between the results of McIntyre[ l] and those of Gummel and Blue[2]. The results can also be reduced to the cases considered by Hines [3] and Tager [4]. The time dependence of the multiplication process has been treated previously by Lee rt a1.[5]. Later Emmons and Lucovsky[6,7] calculated the frequency response of the avalanche current gain for p-i-n diode with spatially constant, but unequal ionization rates. Experimental observation of this dependence has been made by Anderson et u/.[8] and Melchior and Lynch[9]. This paper extends the results of previous authors and derives an analytical expression for mean square avalanche noise current. The theory for multiplication noise in uniform diodes derived by McIntyre[l] considers the randomness of the multiplication process as THIS
PAPER
*This work was initiated while author with Cornell University, New York.
was associated 19
20
I. M. NAQVI
Gummel and Blue[2] have given a generalized noise theory for IMPATT diodes. Their theory considers unequal ionization rates and velocities of electrons and holes and includes the space-charge and transit-time effects in a self-consistent manner. They have shown that their noise expression could be reduced to the results obtained by Hines[3] for the special case of equal ionization rates and velocities of holes and electrons. The objectives of Gummel and Blue[2] and McIntyre [ I] were somewhat different but they have used the same basic assumptions concerning the spectral distribution of noise associated with the ionizing collisions. Despite the similarity of their basic assumptions, it is observed that the low-frequency limit of the results of Gummel and Blue [2] apparently does not agree with the results of McIntyre[ I]. This paper analyzes precisely this problem and derives the correspondence which is expected from the similarity of their basic assumptions and also shows that the results could be reduced to the cases considered by Hines[3] andTager[4]. In this paper, an expression for the mean square avalanche noise current due to the noise generated by the randomness of the multiplication process in the avalanche region of the diode is derived. In order to keep the analysis general, we consider unequal ionization rates and velocities of electrons and holes. The effects of the space charge of the moving charge carriers in the drift region on the electric field in the avalanche region and the transit-time effects are neglected. The neglect of these effects is justifiable at frequencies well below transit-time frequency where these effects do not become very significant. The analysis is therefore limited to the noise generated in the multiplication process in the avalanche region. Hines[3] has shown that the noise generated in the avalanche region can be calculated separately and the effects of space charge and transit time through the drift region carrbe included as an additional factor to obtain the total noise of the diode. However, at frequencies well below the transit-time frequency (o Q L’/M~), this factor reduces to unity. Therefore, in this study, the approximation to neglect these effects is reasonable. The experimental results described in Section 5 have shown that, even at low frequencies, the importance of the factor resulting from consideration of the time dependence of the multiplication process cannot be ignored. Experimental results on
very uniform silicon avalanche diodes are found to be in good agreement with the theory presented here. 2. THE
MODEL
Consider a reverse bias p-n junction having a depletion region of width W. The exact shape of the electric field is arbitrary provided it is sufficiently high in some part of the depletion region to cause impact ionization of carriers. This high electric field region is referred to as the avalanche region or multiplication region of width x, where x, < M’(see Fig. I). The electric field in the avalanche region in
1 ‘.
; I 4 ! .r?
-..: HOLES
d
u
I I
0
-
i
’
‘.
.’
,’
Jp(x,)-J-J,,
,’ x’ t-
-ELECTRONS
I I
* x+dx
XI
AVALANCHE REGION
_
DEPLETION
_
REGION
Fig. 1. Schematic showing boundary conditions
electron and hole flow and the for the avalanche region.
silicon diodes is sufficiently high (> lo” V/cm). so that it is reasonable to make the approximation that the electron and hole velocities, U, and c,, respectively, are constant and at their respective saturated values (v,= 1.05X 107cm/sec[13], c,=7,5X IO” cm/sec[ 141, in silicon). The multiplication of carriers takes place when the electric field in the depletion region exceeds some critical field. The multiplication process may be initiated by injected saturation current across the boundaries of the avalanche region or thermally and/or optically generated carriers within the avalanche regicn. In either case, an electron (or a hole) which is accelerated by the electric field gains sufficient energy to remove an electron from its covalent bond, thus creating an electron-hole pair. A charged particle may create zero, one. two or more electron-hole pairs in a series of ionizing collisions. Each newly created electron and hole traveling in opposite directions may again create zero, one, two or more electron-hole pairs, until the boundaries of the avalanche region are reached.
MULTIPLICATION
The avalanche
current fluctuations
this randomness of the electron-hole
PROCESS
are caused by pair genera-
tion. The dependence of current fluctuations on the multiplication process is expressed in the most direct way in terms of the multiplication factor M. This factor M is a function of the ionization rates of electrons and holes[5] which are dependent on the electric field. In analyzing the behavior of the avalanche diode, both d.c. and a.c. electric fields are determined uniquely from the continuity equations and Poisson’s equation plus the boundary conditions determined by the diode structure and the level of operation. Therefore, from the steadystate solution, there exists a precise value of M for a given level of operation. In the same manner, from the solution of the time-dependent continuity equations, there exists a precise value of M which depends on the frequency of operation, 3. A.C. MULTIPLICATION FREQUENCY SPACE
IN
In this section, an expression for the a.c. multiplication factor M(x, o) is derived by solving the time-dependent continuity equations for the motion of electrons and holes in the avalanche region. M(x) is defined as the average total number of electron-hole pairs generated in the avalanche region due to a single pair generated at some plane x within the avalanche region (Fig. 1). The time dependence of the multiplication process arises from the fact that there is a significant time interval between the initial generation of carriers at a plane x and the collection of all the multiplied carriers at the boundaries of the avalanche region. The time dependence of the multiplication process is governed by the continuity equations. In this section we derive this dependence and show its relation to the intrinsic response time of the carriers. The junctions used in this study are assumed to have uniform breakdown; therefore, the analysis can be reduced to one dimension. The uniformity of the junctions, used for experimental study, has been checked by the method described earlier [ 121. In order to define M(x, w) mathematically, consider that there is a generation of carriers at a plane x within the avalanche region, by some external injection such as a light source, and assume that injected saturation current .I, at the boundaries of the avalanche region is zero. We write the volume
AND
AVALANCHE
21
NOISE
generation rate in terms of a spatial delta-function 6(x’-x),suchthat, g(x’) = SG(x’)S(x’-x)
(1)
where SC(x) is the generation rate of charged particles per cm2 in an element dx at plane x, and x’ is the spatial variable in the avalanche region [0, x,] _ We could also define 6G (x) by integrating equation (I), such that I,Z’g(x’)
dx’ = Jd;‘8G(x’)S(x’-x)
dx’ = SC(x). (la)
This spatial delta-function generation at plane x gives rise to a particle current* at the boundaries of the avalanche region due to the multiplication process throughout the region. This particle current due to the generation at plane x is denoted by 6J(x, 0). An expression for&/(x, w) is obtained by multiplying equation (1) by M (x’, w) and integrating from 0 to x1. 6J(x,o)
= I,x’ ~G(x’)~(x’-x)M(x’,w) = sG(x)M(x,
w).
dx’ (2)
The x-dependence in the expression for &J(x, w) implies that 6J(x, w) is the particle current due to initial generation of electron-hole pairs at a plane x and does not include any other spatial dependence of this current in the avalanche region. From equation (2) M(x. w) can be defined as, M(x,o)
=$$$.
(3)
In writing the continuity equations, it is assumed that recombination of carriers can be neglected since the lifetimes of the carriers (>, lo-* set) are much greater than the transit time through the avalanche region (- 10mioset). The field across the junction is also large enough (> lo5 V/cm) such that the diffusion currents can be neglected in comparison with the drift currents. The time-dependent continuity equations for the motion of electrons and holes in the avalanche *The term ‘current’ is often used for brevity, although the implication is that of ‘current density’. The letter J is used for current density, whereas the letter I is used for current.
22
I. M. NAQVl
region shown ing form
in Fig. 1 can be written
1 alJ,l
aJ,
--+%= u,> at
aJn ad
1 alJ,l ----= L‘, at
in the follow-
~IJ,~I+~IJ~I +tg
(4)
alJ,l +pIJyI +g
(5)
where J,, and J, are the electron and hole current densities; P, and z.~ are the absolute values of the electron and hole saturated velocities; LYand /3 are the ionization rates for electrons and holes, respectively; g is the thermal and/or optical generation rate. The procedure for solving these equations is similar to that used by Lee et a/.[ 151. The boundary conditions are shown in Fig. 1. Combining equations (4) and (5) yields a first-order differential equation in time. This differential equation was solved by Lee et al. for the photomultiplier case by assuming a solution of the form of e’“‘. The validity of the assumptions leading to the first-order differential equations has been justified previously [ 151. In the present context, we are interested in a solution of the continuity equations when the initial generation of electron-hole pairs takes place at plane .Y. From this solution, we can then obtain an expression for the multiplication factor M (x, o) by using the definition given in equation (3). Following Lee et (I/.[ 151 we assume a multiplicative time factor. @‘. for electron and hole saturation currents J,s,, and J,v, and the generation term g(x, t). Then, by eliminating the factor erw’ from both sides of the solution, we obtain SJ(x,w) where M(x) defined as M(x)
E
=
is the d.c. multiplication
(6)
SJ (x)
e
-
factor
and is
From equations product Mu’ writing
I-1:”
el:’ v
tarfi1d.r”
ne-];,’
response
M(x)
SC(x)
1 +iWM(X)T’(X)’
e-l,‘.’ ,I
dx’
time of carriers
(3),
(9)
(7) and (8) it is observed that the is independent ofx. Therefore,
M(X)T’(X)
= M7’
(IO)
we obtain
M(x)
M(x, w) =
(I 1)
I + ~wMT’
The meaning of M and T’ are explained in equations ( 19) and (20) in conjunction with equation ( 12). In solving these equations, the effects of carrierinduced displacement current have been neglected. It has been shown by Kuvas [ 161 that the inclusion of these effects results in modification of the intrinsic response time of the carriers without essentially altering the form of the first-order differential equation obtained by combining these two equations. The correction to the intrinsic response time due to the effects of carrier induced displacement current as obtained by Ku&[ 161 shows that the corrected response time K7’
7=
(12)
where K is a factor that depends on the ionization rates and velocities of the charge carriers and the diode structure. The exact expressions for K are given by Ku&[ 161, however, we give the numeriCal values of K for some simpler cases. For silicon diode having a p-i-n structure, assuming p/cy - 0. I and L’,,/ L’~ - 2 we find that K =
For equal ionization trons and holes,
0.5.
rates
and velocities
of elec-
at
,p,
+u !I
(cr-fildr”
of M(x, W) from equation
SJ (x, 0)
=-=
(7)
and r’(x), the intrinsic plane x, is defined as =
M(x,w)
J: (
-=
6G (x)
T’(X)
SG(X)M(X) 1 +iwM(x)r’(x)
Using the definition we obtain,
I ,)
(,,@dr”&‘.
(8)
This is in agreement with Emmons and Lucovsky[6,7].
the earlier results Writing 7 instead
of of
MULTIPLICATION
PROCESS AND AVALANCHE
MJ, = M,J,,
T’ , in equation ( 11) we obtain
23
NOISE + MpJ.pp
(19)
and (13)
M(x,w) = *.
(20)
From equation (7), two other definitions are obtained. If the avalanche process is initiated by pure electron injected current, the junction multiplication is given by M, = M(x,)
= [I -JOT’ ,,);
(14)
(a-P,J’.dx’j
Similarly, if the avalanche process is initiated by pure hole injected current, the junction multiplication is given by M, = M(0)
=
(15)
1 These definitions of M,, and M, were also obtained earlier by Lee et a1.[5] in slightly different form. From equations (8) and (12), the corrected intrinsic response times of electrons and holes are defined, respectively, as
and
An important conclusion to be drawn from these definitions is that the product of the multiplication factor and the intrinsic response time is independent of the type of the carriers and the ratio of electrons and holes exciting the avalanche process, or that Mprp = Mar,, = Mr.
(18)
It is also significant to realize that the equality in equation (18) has been possible because the correction factor K, due to the carrier induced displacement current, is independent of the type of carriers exciting the avalanche process. M and r, given in equation (18), are defined by the following expressions,
The expression for M(x, o) as given by equation (13) is used in Section 4 to derive an expression for the mean square avalanche noise current.
4. MEAN SQUARE AVALANCHE NOISE CURRENT
In this section, an expression for the mean square avalanche noise current is derived. Near avalanche breakdown, thermal fluctuation (due to diode resistance of a few hundred ohms) are negligible compared to the multiplication noise (which is found to be proportional to M”). The effects of generation-recombination noise and flicker noise are also neglected in this analysis since these effects are proportional to l/frequency and occur at frequencies less than 10 kHz. The noise generated in the avalanche region is assumed to have shot noise CharacteristicsI 17, 181. McIntyre’s [ 1] approach to the problem of calculating the noise current is described below and then extended to include the time dependence of the multiplication process. McIntyre considers the noise generated in an incremental element dx at a plane x within the avalanche region (see Fig. 1). Then by integrating over the entire avalanche region, and including the terms due to injected saturation currents. he finds an expression for the mean square avalanche noise current of the diode. The spectral density of the avalanche noise current due to the generation of carriers in an element dx at plane x is written as [ 1] d&,(x) = 2eMZ(x)A
dJ,,(x)
(21)
M(x) is the average d.c. multiplication factor due to the generation at plane x. J,,(x) is the hole current density at plane x in the avalanche region, and A is the diode cross-section area. The
where
effects of the randomness of the multiplication of carriers in the avalanche region is taken into account by considering the average increase in the hole current dJ,(x) [or the electron current dJ,,(x)], within the element dx, due to the generation of carriers by the electron and hole currents and by the thermally and/or optically generated carriers.
24
I.
This increase in the hole current in the element is given by the continuity equation
dJ,(x)
M. NAQVI dx
(22)
(28)
Only the increase in the hole (or electron) current needs to be considered, since electrons and holes are always produced in pairs. The increase in the electron current in the element dx is given by the following continuity equation, which differs from equation (22) only in sign because the direction of electron flow is opposite to that of hole flow, or that
which differs from McIntyre’s expression by the multiplicative factor (1 + o~M~T~)-‘. For the simple case of cx = p and v,, = u,, = L‘. equation (28) reduces to
dx
=~IJn(X)/+tlJn(x)I+~(x).
-~=al~.(x)i+~~~U(x)~+n(x).
(23)
Equation (21) can be easily extended to include the time dependence of the multiplication process by replacing M(x) by M(x,w) and using the expression for M(x,o) as obtained in the previous section [equation (13)]. The mean square avalanche noise current of the diode in a bandwidth B, due to electron-hole pair generation in an element dx is given as d(P)sval.
= 2elM(x,o)llAdJ,,(x)B.
( i2)a,,al =
,+&M2
i
+
2eAJ,T,M,,“B [,_(,-ki(!$LjZ] 1+ 02MZr” gM’ dx
I,:’ gM dx].
(30)
J,,IM,(w)12+J,s,lM,(w)l’ (25)
+
(29)
where M = M, = M, is the average multiplication factor,& = J,,, +J,,+ I:;1 K dx is the injected saturation current, T, = x,/v is the transit time through the avalanche region, and K is the correction factor due to the carrier induced displacement current. This equation is similar to the one obtained by Tager[4] except for the correction factor K in the intrinsic response time. For the case of @(Emax ) = &a (E,,, ) equation (28) results in
+ k”;‘_;r” =2eAB
K+2 ( j
(24)
The mean square avalanche noise current of the diode, when the generation takes place throughout the avalanche region, can be obtained by integrating equation (24) from 0 to x, and adding the terms due to the hole and electron current iniected into the avalanche region, (i’),,,,
2eAJ,M”B
%$
IM(x,W)lidx]
M,,(w)
= $&
where
(26)
and
(27)
Substituting equations tion (25) we obtain
( 13), (26) and (27) in equa-
Since this expression for the mean square avalanche noise current includes the time dependence of the multiplication process, it is more general than the expression for noise derived by McIntyre1 I]. Figure 2 shows a normalized plot of equation (30) neglecting the generation term R for specific values of ,f‘. T,,, T, and k stated in the figure. The ordinate is 10 times the logarithm of mean square normalized to the full shot noise current, (i’),,,,. noise” of the diode (at M = I), multiplied by M:‘. ‘.Full shot noise in a bandwidth B and at unity multiplication is given by 2~l,B, where I, is the saturation current in the illuminated part of the diode.
MULTIPLICATION
PROCESS
AND
AVALANCHE
25
NOISE
7;p*4XIO-“SEC _,S _ 7;“.4X10-“SEC
10
IO’
IO’ MULTIPLICATION
FACTOR
IO’
M
Fig. 2. Theoretical curves for multiplication noise normalized to full shot noise times W, for various fractions of electron and hole currents.
Different curves correspond to the various ions of electron and hole saturation currents.
frac-
5. EXPERIMENT AND RESULTS The measurements of the avalanche noise were made on extremely uniform p+-n-n+ silicon diodes. In using non-uniform diodes it was found that the total diode noise could considerably exceed the multiplication noise predicted by equation (30), due to the presence of microplasmas and other inhomogeneities in the diode material. In the method described below, the noise is measured in a very small area of the diode, and the background noise from the remainder of the diode is kept low by using very uniformly diffused diode material. The uniform diodes were prepared by vacuum diffusion techniques. A heavy diffusion of boron was made into an epitaxially grown n-layer on n+ substrate. Mesas were formed by chemical etching and the surface was stabilized by further chemical treatment. The uniformity of the diodes was evaluated by plotting profiles of multiplication factors to within a resolution of 2 x lo4 cm [ 121. The multiplication noise was measured in a limited area of 4 x 1OmR cm* of the diode to assure extreme uniformity of the material. This was accomplished by optically exciting the diode into avalanche by using a He-Ne laser. The effects of
background noise, generated in the section of the diode not illuminated by the laser light, were subtracted from the noise measurements of the whole diode including the illuminated area. The experimental setup used is shown in Fig. 3 [ 121. The laser beam is focussed through a microscope to a spot approximately 2 X 10e4 cm in diameter on the p+ surface of the diode. Multiplication noise is measured at 30 MHz using a low noise, high gain 30 MHz amplifier and detector. The bandwidth of the noise measuring equipment is O-5 MHz. Simultaneous measurements of multiplication factor are made by chopping the light at 450 Hz and measuring the photocurrent with a phase-sensitive lock-in amplifier. The results of these noise measurements on a 125 X lo-“ cm diameter mesa p+-n-n+ silicon diode are plotted as a function of multiplication factor in Fig. 4. The ordinate shown in Fig. 4 is 10 times logarithm of mean square noise current, (i2)aVal, normalized to the full shot noise of the diode (at M = l), multiplied by M3. The two experimental curves correspond to an order of magnitude difference in the intensities of the injected light. The intensity of light is indicated in terms of J,,, the photo-injected current density at unity multiplication. The theoretical curve is obtained by determining,
26
I.
SYNC.
M.
NAQVI
SIGNAL MAGNETIC
l,,n,,
SHIELDING
MICROSCOPE
Fig.
3. Experimental
setup for multiplication
and noise
measurements.
Fig. 4. Theoretical and experimental curves for multiplication noise normalized to full shot noise times W. Theoretical curve is for,f’= 30 MHz. T,, = 4 x IO- I-’ set and 7,, = 3.2 x IO ‘I set and X = 0.12. Experimental curves represent the noise generated within a 2 x IO-‘cm diameter spot on a I25 x lo-* cm diameter silicon mesa diode. J,, is the photo-injected current density at unity multiplication.
from the geometry of the diode structure and the absorption coefficient of 6328 A light in silicon, the percentage of light absorbed in the p+-region and the percentage absorbed in the n-region. It is assumed that the thickness of the avalanche region is negligibly small. The photons absorbed in the p+-region excite electron current and the photons absorbed in the n-region excite hole current. In this diode, it was found that 64 per cent of the saturation current is due to holes and 36 per cent of the
saturation current is due to electrons. The theoretical curve in Fig. 4 is based on these fractions of electron and hole currents. Other values used in calculating the theoretical curve are: X = 0.12. T,, = 4 X IO-” sec. 7P = 3.2 X lo-” sec.* The experimental curves show good agreement with the .‘These values are based on electric tained from capacitance measurements area diode.
field profile obon similar large
MULTIPLICATION
PROCESS
theory. The small deviation from the theory at higher current densities is believed to be due to the space charge effects, which were neglected in this analysis. At higher current densities, the results of measurement may also be somewhat affected by the differential diode resistance, which was calculated by McIntyre [ 11to be Rd =
adr)]-‘.
ILI,AM+(/(
(31)
The effect of diode resistance and load resistance have also been considered empirically by Melchior and Lynch [9]. 6.
SUMMARY
AND CONCLUSIONS
Although McIntyre [ l] has correctly formulated the noise mechanism based on the multiplication process, his approximation which neglects the time dependence of the multiplication process, is equivalent to the assumption that w2M%” 6 1 and leads to an incorrect result whenever w’M~T~ 3 1. It can be seen from Fig. 4, that for frequency of 30 MHz, McIntyre’s theory gives incorrect results. if multiplication factor M is larger than 100. This is a direct consequence of the neglect of time dependence of the multiplication process in his analysis. Therefore. the importance of the term 02MZ~’ cannot be ignored even at low frequencies. The noise expression derived in this paper can be reduced to the results obtained by Hines[3] and Tager[4] for the case of equal ionization rates and velocities
of electrons
and holes.
In Tager’s
noise
expression the intrinsic response time is equal to half the transit time of the carriers through the avalanche region Ta. This agrees with our results if we neglect the factor, K, due to the carrier induced disulacement current. Hines[3], on the other hand, arrived at the following expression for the mean square noise current in the avalanche region (assuming equal ionization rates and velocities of holes and electrons and neglecting the generation term 8) (i’) nines = 2elB
(32)
WZTZ2
where I is the total current through the diode and is the time between two ionization events. Hines [ 31 has made two major approximations in deriving
7,
AND
AVALANCHE
NOISE
27
his noise expression. First, he has used T,, the time between two ionization events, whereas, it should be the intrinsic response time r, as shown in the derivation of equation (30). Second, he has neglected the saturation current in solving his current equation; in other words, he assumes h4 to be infinite. This results in the omission of Me2 which would be added to the term in the denominator of his expression, if the saturation current was not neglected. This is equivalent to the assumption that 02M2~2* 1. Equation (30) can be reduced to Hines’ results for the case of equal ionization rates and velocities of electrons and holes, using the limit w’MV 9 1, and replacing r by 7,. By adding M-" to the term in the denominator, it is also observed that the pole at zero frequency in Hines’ expression disappears even without considering any external resistance and space charge effects as mentioned by Hines. Gummel and Blue[2] have shown that their theory can be reduced to Hines’[3] theory for the special case of equal ionization rates and velocities of electrons and holes. They find that the time between two ionization events, T,, used by Hines should be set equal to half the transit time through the avalanche region, 7,. This is in agreement with our results if we neglect the factor due to the carrier induced displacement current. The comparison with Hines’s theory suggests that the theory of Gummel and Blue is also valid in the limit of W’M’T’
%
1.
- The author is deeply indebted to Prof. C. A. Lee and Prof. G. C. Dalman of Cornell University for many valuable technical discussions of the
Acknon,ledRrrnenfs
subject and helpful comments and suggestions.
REFERENCES
1. R. J. Mclntvre. IEEE Trcrns Electrm Devices ED-13. 164, (1966): 2. H. K. Gummel and J. L. Blue. IEEE Trtrns Ek~ron Devices ED-14,569, (1967). 3. M. E. Hines, IEEE Tmns Electron Drcices ED-13.
158.(1966). 4. A. S. Tager, Soviet Phys. Solid-St. 6. 19 19, ( 1965). 5. C. A. Lee, R. A. Logan. R. L. Batdorf. J. J. Kleimack and W. Wiegmann. Phys. Rev. A134,761, (1964). 6. R. B. Emmons and G. Lucovsky. fEEE Trms. Electron Devices ED-13, 297, (1966). 7. R. B. Emm0ns.J. cwnl. Phvs. 38.3705. (1967). 8. L. K. Anderson, P.‘G. McMullin, L. A. D’Asaroand A. Goetzberger. Appl. Phys. Leti. 6.62. (1965).
9. H. Melchior and W. T. Lynch. IEEE Trrrns. Eleclron Devices
ED-13.829, (1966).
28
I. M. NAQVI Melchior and L. K. Anderson, Internutionol Electron Devices Meeting, Washington, D.C. (1965). R. D. Baertsch, IEEE Tram Electron Devices
IO. H. Il.
ED-13.383, (1966); ED-13,987, (1966). 12. I. M. Naqvi, G. C. Dalman and C. A. Lee, 1968 Solid Store Device Research Conference, University of Colorado ( 1968). I. M. Naqvi, G. C. Dalman, and C. A. Lee. Proc.
IEEE, 56,205 1, ( 1968). 13. C. Y. Duh and J. L. Molt, IEEE Tram Elecfron Devices ED-14,46, (1967). 14. V. Rodreiguez and H. Ruegg, /EEE Truns Electron Devices ED-14,44. (1967).
15. C. A. Lee, R. L. Batdorf, W. Wiegmann and G. Kaminsky,J. uppl. Phys. 38,2787, (1967). 16. R. L. Kuvas, Tech. Rep. RADC-TR-68-571. Rome Air Development Center. Griffiss AFB, N.Y. (1969): R. L. Ku& and C. A. Lee, J. crppl. Phvs. 41. 1743, (1970). 17. A. van der Ziel, Fluciuution Phenomenu in Srmiconduciors, Chapter 3, Academic Press. New York ( 1959). S. 0. Rice. Syst. tech. J. (1944); ( 1945). Also, reprinted in Selected Papers on Noise und Stochastic Processes (edited by N. Wax). Dover Publications. New York (1954).
of Electrical (Received
Engineering,
University
of Hawaii,
13 April 1972; in recisedform
Honolulu,
Hawaii,
U.S.A.
12 June 1972)
Abstract-Theoretical and experimental studies of noise generated due to the randomness of the multiplication process in the avalanche region of a uniform diode are presented. The theory extends the results of McIntyre to include the time dependence of the multiplication process. It also shows the correspondence between the results of MCIntyre, Gummel and Blue, Hines and Tager. The spacecharge feedback and transit-time effects have been neglected in this analysis. The theoretical and the experimental results described have shown that even at frequencies well below transit-time frequency, the importance of the factor resulting from consideration of the time dependence of the multiplication process cannot be ignored. The measurements of the avalanche noise on uniform p+--n--n+ silicon diodes are found to be in good agreement with the theory presented here.
1. BACKGROUND
the mechanism for noise generation. In his analysis, McIntyre has assumed that under the lowfrequency limits (frequencies well below transittime frequency), only the d.c. multiplication factor M need be considered. The theory presented in this paper shows that this assumption is justifiable only at low multiplications when o*M*? -=s 1, where o is the frequency, and 7 is the intrinsic response time of carriers. The experimental results showing the agreement with McIntyre’s theory at low multiplication have been reported by several authors [ lo- 121. Hines’[3] expression for noise in the avalanche region is derived by qualitative reasoning. It contains the very simplified assumptions of equal ionization rates and velocities of holes and electrons. The application of this expression to silicon diodes, therefore, suffers from these approximations. The expression for noise derived by Tager [4] also considers the assumptions of equal ionization rates and velocities of electrons and holes. His expression, however, does include the time dependence of the multiplication process. The intrinsic response time obtained by Tager under the above crssumptions is different from those derived in this paper only by the factor arising from the carrier induced displacement current.
describes new theoretical and experimental studies of noise generated due to the multiplication process in the avalanche region of uniform diodes. It extends the theoretical results of McIntyre[ l] by including the time dependence of the multiplication process in the avalanche region. It also shows the correspondence between the results of McIntyre[ l] and those of Gummel and Blue[2]. The results can also be reduced to the cases considered by Hines [3] and Tager [4]. The time dependence of the multiplication process has been treated previously by Lee rt a1.[5]. Later Emmons and Lucovsky[6,7] calculated the frequency response of the avalanche current gain for p-i-n diode with spatially constant, but unequal ionization rates. Experimental observation of this dependence has been made by Anderson et u/.[8] and Melchior and Lynch[9]. This paper extends the results of previous authors and derives an analytical expression for mean square avalanche noise current. The theory for multiplication noise in uniform diodes derived by McIntyre[l] considers the randomness of the multiplication process as THIS
PAPER
*This work was initiated while author with Cornell University, New York.
was associated 19
20
I. M. NAQVI
Gummel and Blue[2] have given a generalized noise theory for IMPATT diodes. Their theory considers unequal ionization rates and velocities of electrons and holes and includes the space-charge and transit-time effects in a self-consistent manner. They have shown that their noise expression could be reduced to the results obtained by Hines[3] for the special case of equal ionization rates and velocities of holes and electrons. The objectives of Gummel and Blue[2] and McIntyre [ I] were somewhat different but they have used the same basic assumptions concerning the spectral distribution of noise associated with the ionizing collisions. Despite the similarity of their basic assumptions, it is observed that the low-frequency limit of the results of Gummel and Blue [2] apparently does not agree with the results of McIntyre[ I]. This paper analyzes precisely this problem and derives the correspondence which is expected from the similarity of their basic assumptions and also shows that the results could be reduced to the cases considered by Hines[3] andTager[4]. In this paper, an expression for the mean square avalanche noise current due to the noise generated by the randomness of the multiplication process in the avalanche region of the diode is derived. In order to keep the analysis general, we consider unequal ionization rates and velocities of electrons and holes. The effects of the space charge of the moving charge carriers in the drift region on the electric field in the avalanche region and the transit-time effects are neglected. The neglect of these effects is justifiable at frequencies well below transit-time frequency where these effects do not become very significant. The analysis is therefore limited to the noise generated in the multiplication process in the avalanche region. Hines[3] has shown that the noise generated in the avalanche region can be calculated separately and the effects of space charge and transit time through the drift region carrbe included as an additional factor to obtain the total noise of the diode. However, at frequencies well below the transit-time frequency (o Q L’/M~), this factor reduces to unity. Therefore, in this study, the approximation to neglect these effects is reasonable. The experimental results described in Section 5 have shown that, even at low frequencies, the importance of the factor resulting from consideration of the time dependence of the multiplication process cannot be ignored. Experimental results on
very uniform silicon avalanche diodes are found to be in good agreement with the theory presented here. 2. THE
MODEL
Consider a reverse bias p-n junction having a depletion region of width W. The exact shape of the electric field is arbitrary provided it is sufficiently high in some part of the depletion region to cause impact ionization of carriers. This high electric field region is referred to as the avalanche region or multiplication region of width x, where x, < M’(see Fig. I). The electric field in the avalanche region in
1 ‘.
; I 4 ! .r?
-..: HOLES
d
u
I I
0
-
i
’
‘.
.’
,’
Jp(x,)-J-J,,
,’ x’ t-
-ELECTRONS
I I
* x+dx
XI
AVALANCHE REGION
_
DEPLETION
_
REGION
Fig. 1. Schematic showing boundary conditions
electron and hole flow and the for the avalanche region.
silicon diodes is sufficiently high (> lo” V/cm). so that it is reasonable to make the approximation that the electron and hole velocities, U, and c,, respectively, are constant and at their respective saturated values (v,= 1.05X 107cm/sec[13], c,=7,5X IO” cm/sec[ 141, in silicon). The multiplication of carriers takes place when the electric field in the depletion region exceeds some critical field. The multiplication process may be initiated by injected saturation current across the boundaries of the avalanche region or thermally and/or optically generated carriers within the avalanche regicn. In either case, an electron (or a hole) which is accelerated by the electric field gains sufficient energy to remove an electron from its covalent bond, thus creating an electron-hole pair. A charged particle may create zero, one. two or more electron-hole pairs in a series of ionizing collisions. Each newly created electron and hole traveling in opposite directions may again create zero, one, two or more electron-hole pairs, until the boundaries of the avalanche region are reached.
MULTIPLICATION
The avalanche
current fluctuations
this randomness of the electron-hole
PROCESS
are caused by pair genera-
tion. The dependence of current fluctuations on the multiplication process is expressed in the most direct way in terms of the multiplication factor M. This factor M is a function of the ionization rates of electrons and holes[5] which are dependent on the electric field. In analyzing the behavior of the avalanche diode, both d.c. and a.c. electric fields are determined uniquely from the continuity equations and Poisson’s equation plus the boundary conditions determined by the diode structure and the level of operation. Therefore, from the steadystate solution, there exists a precise value of M for a given level of operation. In the same manner, from the solution of the time-dependent continuity equations, there exists a precise value of M which depends on the frequency of operation, 3. A.C. MULTIPLICATION FREQUENCY SPACE
IN
In this section, an expression for the a.c. multiplication factor M(x, o) is derived by solving the time-dependent continuity equations for the motion of electrons and holes in the avalanche region. M(x) is defined as the average total number of electron-hole pairs generated in the avalanche region due to a single pair generated at some plane x within the avalanche region (Fig. 1). The time dependence of the multiplication process arises from the fact that there is a significant time interval between the initial generation of carriers at a plane x and the collection of all the multiplied carriers at the boundaries of the avalanche region. The time dependence of the multiplication process is governed by the continuity equations. In this section we derive this dependence and show its relation to the intrinsic response time of the carriers. The junctions used in this study are assumed to have uniform breakdown; therefore, the analysis can be reduced to one dimension. The uniformity of the junctions, used for experimental study, has been checked by the method described earlier [ 121. In order to define M(x, w) mathematically, consider that there is a generation of carriers at a plane x within the avalanche region, by some external injection such as a light source, and assume that injected saturation current .I, at the boundaries of the avalanche region is zero. We write the volume
AND
AVALANCHE
21
NOISE
generation rate in terms of a spatial delta-function 6(x’-x),suchthat, g(x’) = SG(x’)S(x’-x)
(1)
where SC(x) is the generation rate of charged particles per cm2 in an element dx at plane x, and x’ is the spatial variable in the avalanche region [0, x,] _ We could also define 6G (x) by integrating equation (I), such that I,Z’g(x’)
dx’ = Jd;‘8G(x’)S(x’-x)
dx’ = SC(x). (la)
This spatial delta-function generation at plane x gives rise to a particle current* at the boundaries of the avalanche region due to the multiplication process throughout the region. This particle current due to the generation at plane x is denoted by 6J(x, 0). An expression for&/(x, w) is obtained by multiplying equation (1) by M (x’, w) and integrating from 0 to x1. 6J(x,o)
= I,x’ ~G(x’)~(x’-x)M(x’,w) = sG(x)M(x,
w).
dx’ (2)
The x-dependence in the expression for &J(x, w) implies that 6J(x, w) is the particle current due to initial generation of electron-hole pairs at a plane x and does not include any other spatial dependence of this current in the avalanche region. From equation (2) M(x. w) can be defined as, M(x,o)
=$$$.
(3)
In writing the continuity equations, it is assumed that recombination of carriers can be neglected since the lifetimes of the carriers (>, lo-* set) are much greater than the transit time through the avalanche region (- 10mioset). The field across the junction is also large enough (> lo5 V/cm) such that the diffusion currents can be neglected in comparison with the drift currents. The time-dependent continuity equations for the motion of electrons and holes in the avalanche *The term ‘current’ is often used for brevity, although the implication is that of ‘current density’. The letter J is used for current density, whereas the letter I is used for current.
22
I. M. NAQVl
region shown ing form
in Fig. 1 can be written
1 alJ,l
aJ,
--+%= u,> at
aJn ad
1 alJ,l ----= L‘, at
in the follow-
~IJ,~I+~IJ~I +tg
(4)
alJ,l +pIJyI +g
(5)
where J,, and J, are the electron and hole current densities; P, and z.~ are the absolute values of the electron and hole saturated velocities; LYand /3 are the ionization rates for electrons and holes, respectively; g is the thermal and/or optical generation rate. The procedure for solving these equations is similar to that used by Lee et a/.[ 151. The boundary conditions are shown in Fig. 1. Combining equations (4) and (5) yields a first-order differential equation in time. This differential equation was solved by Lee et al. for the photomultiplier case by assuming a solution of the form of e’“‘. The validity of the assumptions leading to the first-order differential equations has been justified previously [ 151. In the present context, we are interested in a solution of the continuity equations when the initial generation of electron-hole pairs takes place at plane .Y. From this solution, we can then obtain an expression for the multiplication factor M (x, o) by using the definition given in equation (3). Following Lee et (I/.[ 151 we assume a multiplicative time factor. @‘. for electron and hole saturation currents J,s,, and J,v, and the generation term g(x, t). Then, by eliminating the factor erw’ from both sides of the solution, we obtain SJ(x,w) where M(x) defined as M(x)
E
=
is the d.c. multiplication
(6)
SJ (x)
e
-
factor
and is
From equations product Mu’ writing
I-1:”
el:’ v
tarfi1d.r”
ne-];,’
response
M(x)
SC(x)
1 +iWM(X)T’(X)’
e-l,‘.’ ,I
dx’
time of carriers
(3),
(9)
(7) and (8) it is observed that the is independent ofx. Therefore,
M(X)T’(X)
= M7’
(IO)
we obtain
M(x)
M(x, w) =
(I 1)
I + ~wMT’
The meaning of M and T’ are explained in equations ( 19) and (20) in conjunction with equation ( 12). In solving these equations, the effects of carrierinduced displacement current have been neglected. It has been shown by Kuvas [ 161 that the inclusion of these effects results in modification of the intrinsic response time of the carriers without essentially altering the form of the first-order differential equation obtained by combining these two equations. The correction to the intrinsic response time due to the effects of carrier induced displacement current as obtained by Ku&[ 161 shows that the corrected response time K7’
7=
(12)
where K is a factor that depends on the ionization rates and velocities of the charge carriers and the diode structure. The exact expressions for K are given by Ku&[ 161, however, we give the numeriCal values of K for some simpler cases. For silicon diode having a p-i-n structure, assuming p/cy - 0. I and L’,,/ L’~ - 2 we find that K =
For equal ionization trons and holes,
0.5.
rates
and velocities
of elec-
at
,p,
+u !I
(cr-fildr”
of M(x, W) from equation
SJ (x, 0)
=-=
(7)
and r’(x), the intrinsic plane x, is defined as =
M(x,w)
J: (
-=
6G (x)
T’(X)
SG(X)M(X) 1 +iwM(x)r’(x)
Using the definition we obtain,
I ,)
(,,@dr”&‘.
(8)
This is in agreement with Emmons and Lucovsky[6,7].
the earlier results Writing 7 instead
of of
MULTIPLICATION
PROCESS AND AVALANCHE
MJ, = M,J,,
T’ , in equation ( 11) we obtain
23
NOISE + MpJ.pp
(19)
and (13)
M(x,w) = *.
(20)
From equation (7), two other definitions are obtained. If the avalanche process is initiated by pure electron injected current, the junction multiplication is given by M, = M(x,)
= [I -JOT’ ,,);
(14)
(a-P,J’.dx’j
Similarly, if the avalanche process is initiated by pure hole injected current, the junction multiplication is given by M, = M(0)
=
(15)
1 These definitions of M,, and M, were also obtained earlier by Lee et a1.[5] in slightly different form. From equations (8) and (12), the corrected intrinsic response times of electrons and holes are defined, respectively, as
and
An important conclusion to be drawn from these definitions is that the product of the multiplication factor and the intrinsic response time is independent of the type of the carriers and the ratio of electrons and holes exciting the avalanche process, or that Mprp = Mar,, = Mr.
(18)
It is also significant to realize that the equality in equation (18) has been possible because the correction factor K, due to the carrier induced displacement current, is independent of the type of carriers exciting the avalanche process. M and r, given in equation (18), are defined by the following expressions,
The expression for M(x, o) as given by equation (13) is used in Section 4 to derive an expression for the mean square avalanche noise current.
4. MEAN SQUARE AVALANCHE NOISE CURRENT
In this section, an expression for the mean square avalanche noise current is derived. Near avalanche breakdown, thermal fluctuation (due to diode resistance of a few hundred ohms) are negligible compared to the multiplication noise (which is found to be proportional to M”). The effects of generation-recombination noise and flicker noise are also neglected in this analysis since these effects are proportional to l/frequency and occur at frequencies less than 10 kHz. The noise generated in the avalanche region is assumed to have shot noise CharacteristicsI 17, 181. McIntyre’s [ 1] approach to the problem of calculating the noise current is described below and then extended to include the time dependence of the multiplication process. McIntyre considers the noise generated in an incremental element dx at a plane x within the avalanche region (see Fig. 1). Then by integrating over the entire avalanche region, and including the terms due to injected saturation currents. he finds an expression for the mean square avalanche noise current of the diode. The spectral density of the avalanche noise current due to the generation of carriers in an element dx at plane x is written as [ 1] d&,(x) = 2eMZ(x)A
dJ,,(x)
(21)
M(x) is the average d.c. multiplication factor due to the generation at plane x. J,,(x) is the hole current density at plane x in the avalanche region, and A is the diode cross-section area. The
where
effects of the randomness of the multiplication of carriers in the avalanche region is taken into account by considering the average increase in the hole current dJ,(x) [or the electron current dJ,,(x)], within the element dx, due to the generation of carriers by the electron and hole currents and by the thermally and/or optically generated carriers.
24
I.
This increase in the hole current in the element is given by the continuity equation
dJ,(x)
M. NAQVI dx
(22)
(28)
Only the increase in the hole (or electron) current needs to be considered, since electrons and holes are always produced in pairs. The increase in the electron current in the element dx is given by the following continuity equation, which differs from equation (22) only in sign because the direction of electron flow is opposite to that of hole flow, or that
which differs from McIntyre’s expression by the multiplicative factor (1 + o~M~T~)-‘. For the simple case of cx = p and v,, = u,, = L‘. equation (28) reduces to
dx
=~IJn(X)/+tlJn(x)I+~(x).
-~=al~.(x)i+~~~U(x)~+n(x).
(23)
Equation (21) can be easily extended to include the time dependence of the multiplication process by replacing M(x) by M(x,w) and using the expression for M(x,o) as obtained in the previous section [equation (13)]. The mean square avalanche noise current of the diode in a bandwidth B, due to electron-hole pair generation in an element dx is given as d(P)sval.
= 2elM(x,o)llAdJ,,(x)B.
( i2)a,,al =
,+&M2
i
+
2eAJ,T,M,,“B [,_(,-ki(!$LjZ] 1+ 02MZr” gM’ dx
I,:’ gM dx].
(30)
J,,IM,(w)12+J,s,lM,(w)l’ (25)
+
(29)
where M = M, = M, is the average multiplication factor,& = J,,, +J,,+ I:;1 K dx is the injected saturation current, T, = x,/v is the transit time through the avalanche region, and K is the correction factor due to the carrier induced displacement current. This equation is similar to the one obtained by Tager[4] except for the correction factor K in the intrinsic response time. For the case of @(Emax ) = &a (E,,, ) equation (28) results in
+ k”;‘_;r” =2eAB
K+2 ( j
(24)
The mean square avalanche noise current of the diode, when the generation takes place throughout the avalanche region, can be obtained by integrating equation (24) from 0 to x, and adding the terms due to the hole and electron current iniected into the avalanche region, (i’),,,,
2eAJ,M”B
%$
IM(x,W)lidx]
M,,(w)
= $&
where
(26)
and
(27)
Substituting equations tion (25) we obtain
( 13), (26) and (27) in equa-
Since this expression for the mean square avalanche noise current includes the time dependence of the multiplication process, it is more general than the expression for noise derived by McIntyre1 I]. Figure 2 shows a normalized plot of equation (30) neglecting the generation term R for specific values of ,f‘. T,,, T, and k stated in the figure. The ordinate is 10 times the logarithm of mean square normalized to the full shot noise current, (i’),,,,. noise” of the diode (at M = I), multiplied by M:‘. ‘.Full shot noise in a bandwidth B and at unity multiplication is given by 2~l,B, where I, is the saturation current in the illuminated part of the diode.
MULTIPLICATION
PROCESS
AND
AVALANCHE
25
NOISE
7;p*4XIO-“SEC _,S _ 7;“.4X10-“SEC
10
IO’
IO’ MULTIPLICATION
FACTOR
IO’
M
Fig. 2. Theoretical curves for multiplication noise normalized to full shot noise times W, for various fractions of electron and hole currents.
Different curves correspond to the various ions of electron and hole saturation currents.
frac-
5. EXPERIMENT AND RESULTS The measurements of the avalanche noise were made on extremely uniform p+-n-n+ silicon diodes. In using non-uniform diodes it was found that the total diode noise could considerably exceed the multiplication noise predicted by equation (30), due to the presence of microplasmas and other inhomogeneities in the diode material. In the method described below, the noise is measured in a very small area of the diode, and the background noise from the remainder of the diode is kept low by using very uniformly diffused diode material. The uniform diodes were prepared by vacuum diffusion techniques. A heavy diffusion of boron was made into an epitaxially grown n-layer on n+ substrate. Mesas were formed by chemical etching and the surface was stabilized by further chemical treatment. The uniformity of the diodes was evaluated by plotting profiles of multiplication factors to within a resolution of 2 x lo4 cm [ 121. The multiplication noise was measured in a limited area of 4 x 1OmR cm* of the diode to assure extreme uniformity of the material. This was accomplished by optically exciting the diode into avalanche by using a He-Ne laser. The effects of
background noise, generated in the section of the diode not illuminated by the laser light, were subtracted from the noise measurements of the whole diode including the illuminated area. The experimental setup used is shown in Fig. 3 [ 121. The laser beam is focussed through a microscope to a spot approximately 2 X 10e4 cm in diameter on the p+ surface of the diode. Multiplication noise is measured at 30 MHz using a low noise, high gain 30 MHz amplifier and detector. The bandwidth of the noise measuring equipment is O-5 MHz. Simultaneous measurements of multiplication factor are made by chopping the light at 450 Hz and measuring the photocurrent with a phase-sensitive lock-in amplifier. The results of these noise measurements on a 125 X lo-“ cm diameter mesa p+-n-n+ silicon diode are plotted as a function of multiplication factor in Fig. 4. The ordinate shown in Fig. 4 is 10 times logarithm of mean square noise current, (i2)aVal, normalized to the full shot noise of the diode (at M = l), multiplied by M3. The two experimental curves correspond to an order of magnitude difference in the intensities of the injected light. The intensity of light is indicated in terms of J,,, the photo-injected current density at unity multiplication. The theoretical curve is obtained by determining,
26
I.
SYNC.
M.
NAQVI
SIGNAL MAGNETIC
l,,n,,
SHIELDING
MICROSCOPE
Fig.
3. Experimental
setup for multiplication
and noise
measurements.
Fig. 4. Theoretical and experimental curves for multiplication noise normalized to full shot noise times W. Theoretical curve is for,f’= 30 MHz. T,, = 4 x IO- I-’ set and 7,, = 3.2 x IO ‘I set and X = 0.12. Experimental curves represent the noise generated within a 2 x IO-‘cm diameter spot on a I25 x lo-* cm diameter silicon mesa diode. J,, is the photo-injected current density at unity multiplication.
from the geometry of the diode structure and the absorption coefficient of 6328 A light in silicon, the percentage of light absorbed in the p+-region and the percentage absorbed in the n-region. It is assumed that the thickness of the avalanche region is negligibly small. The photons absorbed in the p+-region excite electron current and the photons absorbed in the n-region excite hole current. In this diode, it was found that 64 per cent of the saturation current is due to holes and 36 per cent of the
saturation current is due to electrons. The theoretical curve in Fig. 4 is based on these fractions of electron and hole currents. Other values used in calculating the theoretical curve are: X = 0.12. T,, = 4 X IO-” sec. 7P = 3.2 X lo-” sec.* The experimental curves show good agreement with the .‘These values are based on electric tained from capacitance measurements area diode.
field profile obon similar large
MULTIPLICATION
PROCESS
theory. The small deviation from the theory at higher current densities is believed to be due to the space charge effects, which were neglected in this analysis. At higher current densities, the results of measurement may also be somewhat affected by the differential diode resistance, which was calculated by McIntyre [ 11to be Rd =
adr)]-‘.
ILI,AM+(/(
(31)
The effect of diode resistance and load resistance have also been considered empirically by Melchior and Lynch [9]. 6.
SUMMARY
AND CONCLUSIONS
Although McIntyre [ l] has correctly formulated the noise mechanism based on the multiplication process, his approximation which neglects the time dependence of the multiplication process, is equivalent to the assumption that w2M%” 6 1 and leads to an incorrect result whenever w’M~T~ 3 1. It can be seen from Fig. 4, that for frequency of 30 MHz, McIntyre’s theory gives incorrect results. if multiplication factor M is larger than 100. This is a direct consequence of the neglect of time dependence of the multiplication process in his analysis. Therefore. the importance of the term 02MZ~’ cannot be ignored even at low frequencies. The noise expression derived in this paper can be reduced to the results obtained by Hines[3] and Tager[4] for the case of equal ionization rates and velocities
of electrons
and holes.
In Tager’s
noise
expression the intrinsic response time is equal to half the transit time of the carriers through the avalanche region Ta. This agrees with our results if we neglect the factor, K, due to the carrier induced disulacement current. Hines[3], on the other hand, arrived at the following expression for the mean square noise current in the avalanche region (assuming equal ionization rates and velocities of holes and electrons and neglecting the generation term 8) (i’) nines = 2elB
(32)
WZTZ2
where I is the total current through the diode and is the time between two ionization events. Hines [ 31 has made two major approximations in deriving
7,
AND
AVALANCHE
NOISE
27
his noise expression. First, he has used T,, the time between two ionization events, whereas, it should be the intrinsic response time r, as shown in the derivation of equation (30). Second, he has neglected the saturation current in solving his current equation; in other words, he assumes h4 to be infinite. This results in the omission of Me2 which would be added to the term in the denominator of his expression, if the saturation current was not neglected. This is equivalent to the assumption that 02M2~2* 1. Equation (30) can be reduced to Hines’ results for the case of equal ionization rates and velocities of electrons and holes, using the limit w’MV 9 1, and replacing r by 7,. By adding M-" to the term in the denominator, it is also observed that the pole at zero frequency in Hines’ expression disappears even without considering any external resistance and space charge effects as mentioned by Hines. Gummel and Blue[2] have shown that their theory can be reduced to Hines’[3] theory for the special case of equal ionization rates and velocities of electrons and holes. They find that the time between two ionization events, T,, used by Hines should be set equal to half the transit time through the avalanche region, 7,. This is in agreement with our results if we neglect the factor due to the carrier induced displacement current. The comparison with Hines’s theory suggests that the theory of Gummel and Blue is also valid in the limit of W’M’T’
%
1.
- The author is deeply indebted to Prof. C. A. Lee and Prof. G. C. Dalman of Cornell University for many valuable technical discussions of the
Acknon,ledRrrnenfs
subject and helpful comments and suggestions.
REFERENCES
1. R. J. Mclntvre. IEEE Trcrns Electrm Devices ED-13. 164, (1966): 2. H. K. Gummel and J. L. Blue. IEEE Trtrns Ek~ron Devices ED-14,569, (1967). 3. M. E. Hines, IEEE Tmns Electron Drcices ED-13.
158.(1966). 4. A. S. Tager, Soviet Phys. Solid-St. 6. 19 19, ( 1965). 5. C. A. Lee, R. A. Logan. R. L. Batdorf. J. J. Kleimack and W. Wiegmann. Phys. Rev. A134,761, (1964). 6. R. B. Emmons and G. Lucovsky. fEEE Trms. Electron Devices ED-13, 297, (1966). 7. R. B. Emm0ns.J. cwnl. Phvs. 38.3705. (1967). 8. L. K. Anderson, P.‘G. McMullin, L. A. D’Asaroand A. Goetzberger. Appl. Phys. Leti. 6.62. (1965).
9. H. Melchior and W. T. Lynch. IEEE Trrrns. Eleclron Devices
ED-13.829, (1966).
28
I. M. NAQVI Melchior and L. K. Anderson, Internutionol Electron Devices Meeting, Washington, D.C. (1965). R. D. Baertsch, IEEE Tram Electron Devices
IO. H. Il.
ED-13.383, (1966); ED-13,987, (1966). 12. I. M. Naqvi, G. C. Dalman and C. A. Lee, 1968 Solid Store Device Research Conference, University of Colorado ( 1968). I. M. Naqvi, G. C. Dalman, and C. A. Lee. Proc.
IEEE, 56,205 1, ( 1968). 13. C. Y. Duh and J. L. Molt, IEEE Tram Elecfron Devices ED-14,46, (1967). 14. V. Rodreiguez and H. Ruegg, /EEE Truns Electron Devices ED-14,44. (1967).
15. C. A. Lee, R. L. Batdorf, W. Wiegmann and G. Kaminsky,J. uppl. Phys. 38,2787, (1967). 16. R. L. Kuvas, Tech. Rep. RADC-TR-68-571. Rome Air Development Center. Griffiss AFB, N.Y. (1969): R. L. Ku& and C. A. Lee, J. crppl. Phvs. 41. 1743, (1970). 17. A. van der Ziel, Fluciuution Phenomenu in Srmiconduciors, Chapter 3, Academic Press. New York ( 1959). S. 0. Rice. Syst. tech. J. (1944); ( 1945). Also, reprinted in Selected Papers on Noise und Stochastic Processes (edited by N. Wax). Dover Publications. New York (1954).