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Noise associated with electron statistics in avalanche photodiodes and emission statistics of [alpha]-particles

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Title:
Noise associated with electron statistics in avalanche photodiodes and emission statistics of [alpha]-particles
Creator:
Gong, Jeng, 1953-
Publication Date:
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English
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vi, 100 leaves : illustrations ; 28 cm

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Subjects / Keywords:
Avalanches ( jstor )
Diodes ( jstor )
Electric current ( jstor )
Electric potential ( jstor )
Electrons ( jstor )
Ionization ( jstor )
Shot noise ( jstor )
Statistical discrepancies ( jstor )
Statistics ( jstor )
Trucks ( jstor )
Diodes, Avalanche ( fast )
Electronic noise ( fast )
Solid state electronics ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 98-99).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jeng Gong.

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NOISE ASSOCIATED WITH ELECTRON STATISTICS
IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF a-PARTICLES








By

JENG GONG


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA


1983












ACKNOWLEDGMENTS


I am deeply indebted to Dr. Carolyn Van Vliet, Dr. Aldert

van der Ziel, and Dr. Peter Handel for their most generous and valuable guidance and assistance during the preparation of this work. I thank Dr. Alan Sutherland, Dr. Eugene Chenette, and Dr. Gys Bosman for their advice and encouragement. Dr. Wiliam Ellis has been most helpful during many phases of this work and his help, especially in discussions of radiation detection techniques, is greatly appreciated.

I also wish to express my appreciation to Dr. R. J. McIntyre and Dr. Dean Schoenfeld for providing RCA diodes and the Argon-ion Laser, respectively. I am particularly indebted to Mrs. S. L. Wang for the preparation of the figures presented in this work.

My deepest gratitude goes to my parents, my wife, and my son who have always had faith, encouragement, and understanding when it was most needed.

This research was supported by AFOSR contract, #82-0226.












TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS ......... ......................... . i.. ii

ABSTRACT ............. ............................. v

CHAPTER I: INTRODUCTION ....... ....................... 1

PART A
NOISE FROM HOLE-INITIATED
PHOTO AVALANCHE PROCESSES

CHAPTER II: THEORY OF NOISE IN AVALANCHE DIODES .... .........5

CHAPTER III: DEVICE DESCRIPTION AND EXPERIMENTAL SET-UP
FOR MEASURING PHOTODIODE NOISE ... ........... ...20

CHAPTER IV: RESULTS FOR HOLE-INITIATED AVALANCHE CURRENTS ..... 29 PART B
EMISSION STATISTICS OF -PARTICLES CHAPTER V: THEORETICAL PERSPECTIVES ..... .............. ... 49

5.1 Handel's Theory ..... .............. ... 49
5.2 The Allan Variance Theorem ............... 52
5.3 Application of the Allan Variance
Theorem to Counting Statistics ...... 54 CHAPTER VI: EXPERIMENTAL SET-UP FOR LONG-TIME COUNTING ..... 57 CHAPTER VII: COUNTING RESULTS ......... .................. 62

7.1 Convergence of the Allan Variance
for N - . ................62
7.2 Results of the Relative Allan
Variance vs. l/T ..... ............. 62
7.3 Results of the Allan Variance vs. T . . .. 66 CHAPTER VIII: DISCUSSION OF RESULTS ..... ................ ...76










CHAPTER IX: APPENDIX:

REFERENCES

BIOGRAPHICAL


CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ..... .................

EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS



SKETCH . . . . . . . . . . . . . . . . . .


Page


87 91 98 100












Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


NOISE ASSOCIATED WITH ELECTRON STATISTICS IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF a-PARTICLES

By

Jeng Gong

August 1983

Chairperson: C. M. Van Vliet
Major Department: Electrical Engineering

It is known that the standard theories of avalanche statistics for two-carrier impact ionization by Tager, McIntyre, and Personick only deal with processes for which the number of possible ionizations is very large. On the contrary, it is believed that in many modern devices the number N of possible ionizations is finite and perhaps even very small (N = 1 - 5).

In 1978, van Vliet and Rucker developed a new theory for-this

small N case, which has been fully confirmed for the electron-initiated ionization process. Part A of this dissertation presents the results of a detailed noise study with hole-initiated avalanche currents which corroborate van Vliet and Rucker's theory for the second time.

Data obtained from extensive measurements on counting techniques for a-particles radioactive decay from 95Am241 are presented in Part B








of this dissertation. These data have shown that the statistics are non-Poissonian for large counting times (order 1,000 minutes) in contrast with the fact that many textbooks cite a-decay as an example for Poisson statistics.

Detailed measurements of the Allan variance indicated a

"flicker floor" due to the presence of 1/f noise in the decay, of 10- This result is in agreement with Handel's quantum 1/f noise theory. If upheld by further measurements, then this would be the first quantitative indication that 1/f noise is caused by emission of long wavelength infraquanta, such as soft photons causing minute inelastic losses in the scattered wave packet.












CHAPTER I
INTRODUCTION


Avalanche multiplication in devices occurs when the free carriers gain enough energy in an electric field so that they can ionize bound carriers upon impact. This phenomenon has been observed in a number of devices, such as reverse biased p-n junctions, impact diodes, Read diodes, FET's, etc. Most authors consider the ionization process as a continuous process [1-6], which means that either the region over which the avalanching occurs is very long, or the number of ionizing collisions per primary carrier transit N is infinite or at least very large.

Ionizations can occur due to electron or hole impact. The

primary carriers responsible for the ionization can be either thermally generated, or arise from tunneling (as in Zener diodes), or stem from light absorption; this is the case in avalanche photodiodes, as considered in these studies.

For present-day small dimension devices, the number of ionizations per carrier transit N is small. In 1978 van Vliet and Rucker used a new statistical approach, referred to as the "method of recurrent generation functions," to develop a new theory which is valid for an arbitrary number of N [7,8]. This theory has been corroborated for electron-initiated ionization process [9].

Measurements with hole-initiated avalanche currentaremade and investigated; the results will be discussed in PartAof this dissertation.









Whereas most of the noise phenomena, like shot noise, thermal noise, and generation-recombination noise, are well understood, 1/f noise remains an enigma. This noise has been observed almost everywhere: semiconductor devices, music, traffic flow, hourglass flow, the frequency of sunspots, the light output of quasars, etc. Because of its universality, some investigators believe that there must be some universal phenomenon operative in all these manifestations [1012].

One of the theories of 1/f noise is the quantum theory, based on the infrared divergence phenomenon and developed by Handel [10]. This theory is fundamental in the sense that it derives the 1/f spectrum from basic quantum physics at the level of a single charged particle subject to scattering, although the final result depends essentially on the presence of many carriers, making 1/f noise similar to diffraction patterns which are one-particle effects but can be seen only if many particles are diffracted. In addition, the theory is universal in the sense that any infraquanta with infrared-divergent coupling to the current carriers will give a contribution to the observed 1/f noise proportional to their coupling constant. Such infraquanta are, for example, very low frequency photons, various types of phonons, shallow electron-hole pairs on the Fermi surface of a metal, spin waves, correlated states [11], etc.

Recently, this theory has been reformulated with quantumoptical terminology and compound-Poisson statistics in a paper written by Van Vliet, van der Ziel, and Handel, which led to the idea of









verifying the theory on a "clean" system outside the domain of solidstate physics-radioactive a-decay [13].

The purpose of this study is to examine the existence of 1/f

noise in radioactive decay from 95Am241 c-particle source. The results are presented in Part B of this dissertation.





























PART A
NOISE FROM HOLE-INITIATED PHOTO AVALANCHE PROCESSES












CHAPTER II
THEORY OF NOISE IN AVALANCHE DIODES The first theories for noise in avalanche diodes were given by Tager (1965) [1] and by McIntyre (1966) [2]. Tager assumed that the ionization coefficients for ionization by electrons and holes are equal. Under these conditions the noise spectral density is given by


Sld = 2q prM3 (2-1) where Id is the avalanche current, Ipr is the primary photocurrent, q is the electronic charge, M is the average gain, and where it is assumed that the incoming particles show full shot noise.

The theory by McIntyre allowed for the possibility that the ionization coefficients are unequal. Let c(x)dx be the probability for ionization by an electron in the interval (x,x+dx) and let (x) be the ionization probability by a hole. For simplicity, McIntyre assumed (x) = kc(x). The noise spectral density is then given by


Sd = 2qIprM3 [ (1 - k)[M --) (2-2)


for the case that the primary carriers injected into the avalanche region are electrons; in case these carriers are holes, replace k by 1/k; if both occur, add the two partial results.









It is not necessary to commit oneself to Poisson statistics of the primary particles; so, one can consider immediately the statistics of the avalanche process. Thus let X denote the offspring (plus original carrier) due to one incoming primary carrier. Then, according to the variance theorem,


SId = M2S + 2q1 var X (2-3)


where M = . From this we find that (2-1) and (2-2) are tantamount to


var X = M2(M- 1) (Tager) (2-4) and


var X = M(M- ) + k(M- 1)2M (McIntyre) (2-5)

The above theories assumed that the region over which avalanche occurs is very long, so that the number N of possible ionizations per carrier transmit is very large. Therefore, these theories can be referred to as asymptotic theories. Lukaszek et al. at the University of Florida (1974) were the first authors to show that this assumption is unrealistic [14], in particular for the region of onset (low gain) of avalanche multiplication. They developed a theory forthe case that N = 1 and N = 2. Besides for short avalanche regions, these formulas should always be applicable at the onset of avalanche ionization. For N = 1, their results give










var X = M(M -1) + 2i (Lukaszek) (2-6)


where X is the a priori chance for ionization by an electron after the electron has gained enough energy to ionize, whereas 11 pertains to ionization by a hole.

In 1978 van Vliet and Rucker reinvestigated the problem. A new theoretical method was developed, named the "method of recurrent generating functions" [7,8]. By this method they were able to solve the complete problem, in which N possible ionizations per carrier transit, or per traveling hole-electron pair (the electron and hole going in opposite directions) are possible. In this case the basic parameters are the a priori ionization probabilities by electron impact (X) or by hole impact (p), once these carriers have covered a path which is long enough to gain the necessary ionization energy from the electrical

field.

For the case that the primary carriers injected into

the avalanche region are electrons, the following results are found


N
e ,e> ( + X)(l - k)
MN N (1 + kX)N I - k(l + X)N+l



MeM )1- k) 1
e <(AXe)2> = N -)(I k) + 1 - kA2 var X - 2 + + k- 1 + k


,[Mek 1 1 (2-8)









The asymptotic limit for N -. is not trivial; in the paper of van Vliet and Rucker it is shown that (2-7) and (2-8) lead to McIntyre's results.
For hole initiated avalanche case, replace k by I/k, X by i, and 1i by X. The results are


h. h -h (1 k)[k(l + ji)]N
MN k (l + _ (k+ )N+l (2-9)


M h(Mh r k (1- 2
var X h <(AX )2> N N � k ) - k) - 2 k + p N N ~2k + pk +


x + I _] (2-10)



We now consider Equations (2-9) and (2-10) in more detail. At the onset of ionization the field is just high enough to sustain one possible ionization in the avalanche region; thus N = 1. With increasing field, the value of p, denoted as 1(l), increases and so the gain increases according to Equation (2-9). The length 1 necessary to gain the ionizing energy simultaneously decreases, until two ionizations per carrier transit are possible. The value of 1(l), just prior to this is denoted as (l) max' When two ionizations per carrier transit are possible, the diode switches over from the regime N = 1 to the regime N = 2. To realize the gain M of that operating point, .the value of i for the new regime, denoted as p(2),is considerably less than the value of i(l)max prior to the switch-over, as is found by inverting (2-9) for 1 with M fixed, taking N = 1 and N = 2,









respectively. See Figure 2-1. At the M value for which the switchover occurs, the variance jumps from var X to var X2. Equation (2-10) indicates that the decrease in p and the increase in N cause var X to make a positive jump at this particular M, which we call a breakpoint value.

If now the field is further increased, the paths Z1 + k2

decreases, p(2) increases, and M increases, until at p = p(2)max three ionizations per carrier transit are possible. Once the regime N = 3 is possible, the -p(3) for these processes is much smaller than 11(2) in max
order to realize the same M. The switch-over marks another breakpoint value for M, at which the variance jumps due to the reduction of 11, from

(2) max to 11(3), etc. We thus obtain an overall curve of var X versus M, which shows discontinuities at the breakpoint values of M, for which the regime switches N N + 1. In order to construct this overall curve we need only plot Equation (2-10) with -p expressed as li(M,N) according to the inversion of (2-9), where N = 1 and a = i(l) up to

(1)max ; next N = 2 and 11 = p(2) up to p(2) max, etc.

We conjecture that p(N)max is related to the pole value,

1p(N), for which the denominator of (2-9) is zero. These pole values are listed in Table 2-1. On general ground we assumed that the p max is related to 1p by


_ N p(N) (2-11) (Nmax N + 5.


The reason for this presumed relationship is that for very large N we must have p(N)max + (N) in order thatM can rise without saturation.
















































Figure 2-1. p vs. M







Table 2-1. hpole(N) and max(N) for different values of k, all values x 10-6


N

k 1 2 3 4 5 6 7 8 9 10

Ppol e
0.02 141,500 57,960 35,050 24,820 19,130 15,520 13,040 11,240 9,868 8,793 0.016 126,500 50,720 30,360 21,370 16,400 13,270 11,130 9,575 8,398 7,476 0.014 118,322 46,807 27,833 19,520 14,944 12,073 10,112 8,692 7,618 6,778 1imax

0.02 23,580 16,560 13,140 11,030 9,565 8,465 7,607 6,917 6,343 5,862 0.016 21,080 14,490 11,390 9,498 8,200 7,238 6,493 5,892 5,399 4,984 0.014 19,720 13,373 10,437 8,676 7,472 6,585 5,899 5,349 4,897 4,519









Converting Equation (2-3) to terms of equivalent saturated diode current, we obtain


Sld = M2slpr + 2qlpr var X = 2qleqd (2-12)


When the primary current shows full shot noise, we have

I Ir =M2 + var X (2-13)



while


Id/Ipr = M (2-14)


Thus, suppose we plot theoretically log (M2 + var X) versus log (M). Such a curve is identical to a plot of log (Ieqd) versus log (Id); the origins of the two curves are displaced by a line under 45', the X-axis and Y-axis displacements being log (I pr). Therefore, by comparing the experimental curve of leqd versus Id on log-log paper with the theoretical curves for M2 + var X versus M on log-log paper, we find immediately I from the displacement of the origin. A special favorable feature pr
is that the theoretical curves of M2 + var X on log-log paper are very close for different values of k. Hence k needs only be approximately known for the determination of I pr. Once Ipr has been found, we then plot leq/I pr vs. Id/I pr on linear paper, and compare it with curves for M2 + var X versus M on linear paper. From the magnitude as well as from the breakpoints in the curve, k is obtained for the best fit. The




13




computer plots for M2 + var X versus M are obtained from (2-9) and (2-10). The results are given in log-log form in Figures 2-2 to 2-4 and on linear paper in Figures 2-5 to 2-7.








k-- 0.0 1 4


1 10


Var X + M2 for k = 0.014 in log-log scale


1001


Figure 2-2.










k= 0.0 1 6


C4

x
010












1
1 10


Var X + M2 vs. M for k = 0.016 in log-log scale


Figure 2-3.









100










+
x a10


k= 0.02


Var X + M2 vs. M for k = 0.02 in log-log scale


1 10


Figure 2-4.


























k= 0.0 14


N: 9


McIntyre


N:8


N: 7


Figure 2-5. Var X + M2 vs. M for k = 0.014 in linear scale




18





















1 k: 0.0 1 6








9



N= 9 7
McIntyre
+
x N: 8




5
N 7 N 6

3

N 5 N-4




1 1.1 1.2 1.3 M







Figure 2-6. Var X + M2 vs. M for k = 0.016 in linear scale
























k =0.02


N:9


McIntyre


N: 7


N:6


N :5


Var X + M2 vs. M for k = 0.02 in linear scale


Figure 2-7.













CHAPTER III
DEVICE DESCRIPTION AND EXPERIMENTAL SETUP FOR MEASURING PHOTO DIODE NOISE


Two specially designed RCA "reach-through" diodes are used

in these hole-initiated avalanche process studies. Figure 3-1 shows the diode structure and impurity concentration profile. This structure is designed such that the depletion region in the p-type material reaches through to the lightly doped i-region when the reverse-bias voltage is about 200 volts. Further increase in the reverse-bias voltage quickly depletes the Tr-region.
+
A window is opened on the n p junction side of the diode so that
+
light can be shone on the n region and then generate electron-hole pairs. Because the diode is reverse biased, holes will drift into the high-field region and initiate avalanche processes. The wavelength of the light is chosen at 502.8 nm so that the absorption depth is shallow, no light can penetrate to the p-type region, which excludes the possibility of electron-initiated avalanche multiplication.

The block diagram of the whole system is shown in Figure 3-2. The light source is a Spectra Physics 171-08 Argon-ion Laser. The TEM-OI mode is used. This laser gives a doughnut-shaped light which is focused by a plain-convex lens. Figure 3-3 shows the biasing circuit of the device under test (DUT). High Q capacitors are used, so as to reduce the thermal noise as much as possible. Six 70-volt batteries and





























-4X 1012


120 /im


15
3.4X 10
n -- T1/2 LD



5 X 1012
1/2
13"' LA


P+= 1 Pm


/4 LD


LD= 1.7 pm LA 7.6 jFm


deep


Figure 3-1. Diode structure and impurity concentration profile
















Device Under Test See Fig. 3-3


Standard Noise Source


Low Noise Amplifier See Fig. 3-5


Figure 3-2. Block diagram of the noise measurement system


500Kfl.








20V
O.,pF O.IF 1oo2 SO2N4220A


NOISE 0 OUTPUT TO LNA


Figure 3-3. Biasing circuit of the DUT









a 5.5 M potentiometer give the correct bias voltage needed. The diode current Id is monitored by a Keithley 602 solid-state electrometer which is capable of measuring current accurately to less than 1 picoampere. An N-channel FET source follower is set in between the biasing circuit and the low-noise amplifier (LNA) to match the impedances.

The 5722 vacuum noise diode is chosen as the standard noise

source because its known level of Gaussian white noise and proven performance. The complete circuit is shown in Figure 3-4. The anode current is adjusted by varying the heater voltage, VHH, with a car battery to supply heater current. The anode load is a 100 millihenry high Q miniature inductor.

The low-noise amplifier as designed by L. M. Rucker of the University of Florida is shown in Figure 3-5; it uses a high transconductance JFET, type 2N6451, as the input of a cascade input stage. The output is a bipolar transistor, type 2N3904. The input stage has an overall gain of 30 dB. The second and third stages are identical 20 dB gain cascades using 2N4220A FET inputs and 2N3904 bipolar outputs. Because the gain of the first stage, no special effect was made to reduce the noise in the later stages. The final stage is a unity gain buffer having an output impedance of approximately 50 ohms. The buffer is connected to one of the earlier stages through a rotary switch having a low capacitance between the contacts. The overall gain of the amplifier may then be set to 30, 50, or 70 dB. The bandwidth depends on the gain setting and is 3.5, 2.5, and 2.2 MHz, respectively.








5722


Noise Diode


12V


Circuit of a noise standard using a vacuum diode


Noise Output


Figure 3-4.












+20V


Circuit for a low-noise amplifier (LNA)


INPUT


Figure 3-5.









Immediately following the LNA is the HP-310A Wave Analyzer,

which is used essentially as a narrow-band, high-gain, tunable filter. It is used in the AM mode with a bandwidth of 3 KHz.

The output of the wave analyzer is connected, through an RC

low-pass filter (or an integrator), to the Y-axis input of the MFE 815 X-Y plotter with an MFE 7T time base on the plotter's X-axis input terminals. The Y-direction displacement is proportional to the output voltage of the HP-310 Wave Analyzer and its time average over a certain period of time can be obtained from this plotter. This time-averaged voltage reading gives more accurate results than that obtained directly from an rms volt meter.

The whole system is first operated with both the light and

calibration noise source off. The variable capacitor (see Figure 3-3) is tuned at the minimum capacitance position and the receiver (HP-310 Wave Analyzer) is set to the resonant frequency of the biasing circuit, 203 KHz, such that the maximum noise output is obtained. A voltage meter reading M1 is recorded and the noise power at the output of the aave analyzer in this case is


M= BGIS bkgJ (3-1)


where B is the bandwidth of the whole system, G the power gain of the amplifier, and Sbkg stands for the spectral density of background noise, which includes the amplifier's noise and noise due to the dark current of the diode.

Next, we turn the calibration noise source on and record the meter reading M2 and the average anode current of the 5722 saturated









vacuum tube diode, IA. Now,


M 2 GB(Sbkg + Scal) (3-2)


where G, B, and Sbkg are the same as in the previous case, Scal = 2qlA is the spectral density of the calibration noise source where q is electronic charge.

Finally, we turn on the light and turn off the calibration noise source and obtain a new reading M3,


M3 = GB(Sbkg + Sld + Slaser) (3-3)



where Sld is the noise spectral density due to avalanche processes, and Slaser is the noise introduced by the laser light source which is considered as constant background noise. Here we assume SId and S laser are independent.

The equivalent saturated diode current for Sld + Slaser, Ieq, can then be obtained by calculating



Ieq = [r 3 M IjIA (3-4)



The procedure is repeated for various bias voltages of the

DUT. Experimental results are plotted and compared with the theory given in Chapter II.












CHAPTER IV
RESULTS FOR HOLE-INITIATED AVALANCHE CURRENTS


Table 4-1 lists the experimental data for an RCA diode. The photocurrent Id generally rises monotonically with the bias voltage Vb. Figure 4-1 shows the measured data of Id/I dVb = 30 V versus Vb. They are in accord with the manufacturer's data which indicates that no electron-initiated avalanche occurred in these experiments. From Chapter II we have


SId = Slaser = 2qIeq (4-1)


Now, let SId = 2 qIeqd, Slaser = 2eqleql, then


'eq = Ieqd + Ieq. (4-2)


If the multiplication factor M equals unity, and the primary photocurrent shows full shot noise, then from Equations (2-10) and (2-12) the following result is obtained:


SId = SIpr = 2ql pr (for M = 1) (4-3)


In this case, leqd = Ipr. The photocurrent Id can hardly be multiplied at the bias voltage of Vb = 30 V; therefore, we can assume Ipr = Id Vb=30V with little error. The constant Ieqz can now be decided by











Table 4-1. Experimental


data for I = 412 nA
pr


V Id leq leqd(leq - 3,733) Id/412 Ieqd/412


30 100

150 170 185 200

240 260 275

300 320 340


412 417 422 432 442 449 458

469 480 492

512 532


4,145 4,159 4,180 4,268 4,480 4,637 4,722 4,792 5,058 5,352 5,887 6,855


412 426 447 535 747

904 989

1,059 1,325

1,619 2,154 3,122


1 .000 1.012 1.024 1 .049 1 .073 1.090

1.112 1.138 1.165

1.194 1.243 1.291


1.000 1.030 1.085 1.299 1.810 2.190 2.400 2.570 3.220 3.930 5.230 7.580
















































0
II .03































Figure 4-I.


Manufacturer's Data


0 Eep. Data


Id/Id Vb=30V


vs. Vb for Ipr = 412 nA









leq = eq Vb:.30V - I' Vb=30V (4-4)


For the data presented in Table 4-1, leqk = 3,733 nA; Ieqd can then be found by using Equation (4-2).

After obtaining leqd' we plot log (Ieqd) versus log (Id). As discussed in Chapter II, the primary current can then be decided by comparing this plot with theoretical curves. The Ipr so determined in Figure 4-2 is exactly equal to what has been assumed, namely Ipr = IdlVb=3OV. The results of leqd, Id/Ipr, and Ieqd/Ipr are also given in Table 4-1.

In the data given here, we show the following types of measurements: (1) the avalanche photocurrent Id as a function of bias voltage;

(2) the noise, as expressed in the equivalent saturated diode current leqd, as a function of bias voltage; (3) leqd as a function of diode current Id. The measurements are typical of many others. We determined the diode current Id and the noise Ieqd for different light intensities. The data for different light intensities are quite similar and generally consistent with each other as predicted by the theory.

Figures 4-3 to 4-5 show the experimental data obtained for a

primary current Ipr of 412 nA. The Id vs. Vb graph shows a slight bend which marks the reach-through voltage. The noise is plotted in Figures 4-4 and 4-5 as a function of Vb and Id, respectively.

Table 4-2 lists the data obtained for a different light intensity. The leqk for these measurements is obtained by Equation (4-4).


I = 3,962 nA - 180 nA = 3,782 nA (4-5)
eq






























-Theoretical Curve for k:0.014

* Exp. Data +
x


4

























Figure 4-2.


I d(nA)


Log(I eq d vs. log(Id)


for I = 412 nA











550








500





Reach Through Point


450
_-o








4001
0 100 200 300
Vb( V) Figure 4-3. Id vs. Vb for Ipr = 412 nA





































I I
vb(v)


100


200


300


400


vs. Vb for Ipr


eq d


4000 V-


3000 I-


2000 F-


1000h


= 412 nA


Figure 4-4.


















3500 3000 2500

4

2000



15001000




500


200
400 450 500 'd (A)







Figure 4-5. leq vs. I d for IDr = 412 nA










Table 4-2. Experimental data for I
pr


V Id leqd(leq - 3,782) Id/180 leqd/180


30 100

135 150 175 200 240 260 280 300 315 330 340 350


180 181 185 190 195

199 204 208 214 220 226 231 237 244


3,962 4,039 4,050 4,104 4,172 4,234 4,309 4,407 4,506 4,608 4,793 4,967 5,217 5,526


180 257 268 322 390 452 527 625 724 826 1,011

1,185 1,435 1,744


= 180 nA


1.000 1.006 1.028 1.056 1.083 1.106 1.133

1.156 1.189 1.222 1.256 1.283 1.317 1.356


1.00 1.43 1.49 1.79 2.17 2.51 2.93 3.47 4.02 4.59 5.62 6.58 7.97 9.69









which is very close to what we found for previous measurements, 3,733 nA,atadifferent light intensity. This is in accord with our assumptions that Slaser is a constant background noise and independent of SId.

By comparing the curve of log (Ieqd) versus log (Id) with the theoretical plots, we note that the curves coincide if the origins are displaced under 450 by an amount log Ipr, with Ipr = 180 nA, see Figure 4-6. Figures 4-7 to 4-9 show the results of Id versus Vb, leqd versus Vb, and leqd versus Id,

Dividing now both axes of Figure 4-5 by the primary current, we can make a plot of the experimental values


leq Ipr M2 + var X versus Id/ I pr = M (4-6)


This result is given in Figure 4-10.

The breakpoint structure is not well pronounced, which is due

to the fact that the current gain, or multiplication factor M, for holeinitiated avalanche current is too small. Therefore, large-voltage increments are necessary in order to observe current increase while increasing the bias voltage, and the breakpoint structure could be easily glossed over. However, we compare the results of Figure 4-10 with the theoretical curves of M2 + var X versus M. The best fit, with respect to magnitude, is obtained for k = 0.014. The experimental data plus the theoretical curve is given in Figure 4-11. Comparison of these experimental results with the asymptotic theory (McIntyre's) would result in a k value of 0.03, which is too large to believe for a modern device.



























I 000








"0


100


/
/
45o


I I I I I I


I I I I I I I I I


10 100 1000

Id (nA)












Figure 4-6. Log(Ieq ) vs. log(Id) for Ipr = 180 nA dp













240-


220-


Reach Through Point


200h-


180 l-


160


100


1.50


200


250


300


350


Vb( v )


Figure 4-7. Id vs. Vb for Ipr = 180 nA











2000 1500







500




0 I I
0 100 200 300 400
v
Fb





Figure 4-8. Ie vs. Vb for Ip = 180 nA















2000 1500


C

1000 500



0
150 200 250 1d1 (nA) Figure 4-9. 1eq vs. I for I pr = 180 nA

















































Figure 4-10.


1.1








Var X + M2 vs. M for Ipr


1.2


1.3


= 412 nA




44















9
Theoretical Curve for k: 0.0 14
/
/
Exp. Results
f

7 /
N: 8


C14



> 5 V N:7

/


N:-6
/
/
3 ,-/NN:
/
N 5


N:4




1 1.1 1.2 1.3










Fiqure 4-11. Var X + M2 vs. M with theoretical curve for I = 412 nA pr









The experimental M 2 + var X versus M curve obtained from lower light intensity measurements is presented in Figure 4-12. Comparison with the theoretical curve for k = 0.014 is made in Figure 4-13. This diode shows consistent noise performance while operated at different light intensities.

From the data presented above for low M values and various

light intensities, we conclude that Van Vliet and Rucker's theory is applicable for hole-initiated avalanche currents.






















9 7



x


5 3








1 1.1 1.2 1.3 M Figure 4-12. Var X + M2 vs. M for I = 180 nA pr
























Theoretical curve for k: 0.014 Exp. Results


/ N:6



4 N:5
N 5


Figure 4-13. Var X + M2 vs. M with theoretical curve for I


I.1 1.2 1.3


= 180 nA






























PART B
EMISSION STATISTICS OF cx-PARTICLES













CHAPTER V
THEORETICAL PERSPECTIVES


5.1 Handel's Theory

It is known that upon scattering a beam of electrons will

emit bremsstrahlung. The power spectrum W(f) of the emitted radiation is independent of frequency (W = constant) at low frequencies and decreases to zero at an upper frequency limit fm which is approximated by E/h, where E is the kinetic energy of the electrons, h the Planck constant. Consequently, the rate of photon emission per unit frequency interval is N(f) = W/Hf, i.e., proportional to 1/f (see Figure 5-1). Therefore, we conclude that the fraction of electrons scattered with energy loss c is proportional to I/:, i.e., the relative squared matrix element for scattering with energy loss E is IbT()12 % 1/6.

If the incoming beam of electrons is described by a wave function exp[(i/f)(Pi - r- Et)], the scattered beam will contain a large nonbremsstrahlung part of amplitude a, and an incoherent mixture of waves of amplitude abT(E) with bremsstrahlung energy loss E ranging from some resolution threshold c0 to an upper limit A < E, of the order of the kinetic energy E of the electrons,



r = exp r - Et]a 1 + f bT()eit/dE] (5-1)
0


where bT()z IbT(E)eiYe has a random phase Y which implies incoherence of all bremsstrahlung parts, and IbT(E)2 is proportional to I/c.

49















W W) N(f) :W hf

W(f)actua I












fmf










Figure 5-1. Spectral density of bremsstrahlung in power W and
photon emission rate N








In Equation (5-1) the frequency-shifted components present in the integral interfere with the elastic term, yielding beats of frequency E/h. The particle density given by Equation (5-1) is



IT12 =a2{1 + 2fA 1bT(E)icos t .YJdE



+ r fA b*(E)bT(e )eiE ' )t/'r-dEd61 (5-2) C0 E0


the second term in large parentheses describes the particle density beats.

If the particle concentration fluctuation is defined by 61 12 = 1 12 - <1 12>, its autocorrelation function will be


22 >_ <2212>2


- 1a14A Ib( )12cos ]dE (5-3)




which is proportional to jb(6)j2 and hence proportional to 1/f. Therefore, the spectral density of the particle concentration fluctuation [the Fourier transform of Equation (5-3)] is proportional to 1/f.
The relative bremsstrahlung rate lb(E)j2 can be derived as follows. The constant spectral energy density can be written as w(f) = 4e2(AV)2/3c3K, where e is electronic charge, c is the velocity of light, K is the dielectric constant of the medium, and AV is the


- I








velocity change in the scattering process. The relative scattering rate density with energy loss �, lb( )l2, is obtained by dividing w(f) by the energy of a photon E = hf:


Ib(f)2 - 4e2(AV)2
3c32 IrtfK


b(s) 12 - 4e2(AV)2
3c hsK


e2 2(A-02 _ cLA h-c 3rrf K fK


aA
K


A - 2 (Z)2
3 'r


e2
and a =
Tic
where K0 is

sity of the


is the fine structure constant. In the MKS system, a the dielectric constant of vacuum. The spectral denrelative fluctuations is


S 2(f)-2 = 2[l + cA kn (A/e0)-2ctA/fK 2'p


= 2aA/f K


5.2 The Allan Variance Theorem

The main link between counting statistics and particle current noise is provided by MacDonald's theorem,


d - 2 Sm (W)W- sin wTdw


(5-8)


where


(5-4) (5-5)


-AV
c


(5-6)


e2 TWcK0


(5-7)









with inversion



Sm(w) = 2w sin T[ M dT (5-9) where is the variance of the total number of particles detected in
T
a time interval (t, t+T) and Sm(w) the noise spectral density of the flux fluctuations Am(t). This theorem is useful for Poissonian statistics. Unfortunately, for 1/f noise, Equation (5-8) is not applicable, since the integral diverges. However, a useful concept in this case is the "two sample variance" or "Allan variance." Let m(l) be the average
T
counting rate in (t, t+T) and M(2) the counting rate in (t + T, t + 2T).
T
Then the Allan variance is defined by

A2 = 1 () T2 (5-10)
GmT = <(mT T


and


A2 = 1 - M(2))2> = T2A2 (5-11)
MT 2 T T mT


The variance [A2 which means ( A ) 2] turns out to be finite for 1/f noise.

The theorem reads [15]



aA2 (T) = 4 Sm ( ) sin d (5-12)


with inversion









.1 iM Cos 1P~
S-() dP o P 2P r(A2P(T)
m 2rif 1 - 2 P OTP MT


(5-13)

For Poissonian shot noise Sm(W) = 2mo, where m is the average counting A2
rate. Substituting Sm(w) into Equation (5-12), one has aMT(T) = moT. For 1/f noise with a spectrum of S0 2C
, = Tw , where C is a constant, the Allan variance cA2 = 2CT2 Zn 2. The various results are summarized in Table 5-1.

5.3 Application of the Allan Variance Theorem to Counting Statistics

The presence of 1/f noise in counting statistics can now be

determined from a measurement of the Allan variance as a function of T. For suppose that the noise is composed of shot noise and 1/f noise, i.e.,


a(T) = m T + 2CT2 Zn 2 (5-14)
M T 0


recall that = m0T, then a measurement of the relative Allan variance R(T) = M(T)/2 yields


R(T) T 2C' kn 2 (5-15)

0


I/m oT is dominant, hence R(T) is proportional to I/T. When T is long enough, 2C' Zn 2 becomes dominant; R(T) is, therefore, a constant, called the "flicker floor."









Table 5-1. Various results of Allan variance theorem


S (W)


Poissonian
shot noise General
shot noise 1/f noise


Lorentzian
flicker noise


Pathological
noise


2m0



2T 2rC/ I wI


4B 2
a +2


= mOT


2
= KmoT 2CT2 kn 2


3 [4e- T - e-2aT a2


L/IwI X-1
0 < X < 4; X # 2


+ 2cT - 3]


LTX(l - 2X-2)
sin (7TX/2)(X + 1)


crA2 (T) MT








/
According to Handel's theory, the constant C' in the 1/f noise term in the rela4ve Allan variance is


C' = 2A /K (5-16) where is a coherence factor; for a-particles it is expected to be close to one; a is the fine structure constant 1/137, and A = 2(Av)2/3c c2 where Av is the velocity change of the particles in the emission process, c is the velocity of light. For a-particles one finds


aA = 8.32 x 10-7 x E (5-17)
K K


where E is energy in MeV and K is the dielectric constant of the radioactive material.

In these experiments E = 5.48 MeV, so that the value of 2C' kn 2 (the value of the flicker floor, F) is


F = 4C 8.32 x 10-7 x 5.48 x kn 2 = 126.4 x 10-x7 x
K (5-18)













CHAPTER VI
EXPERIMENTAL SET-UP FOR LONG-TIME COUNTING


The block diagram of the counting system being used to investigate 1/f fluctuations in the a-particle emission rate is shown in Figure 6-1. The source is 95Am241, which decays with a half-life time of T1/2 = 458 years with the emission of 5.48 MeV a-particles into 93NP237 The detector, a silicon surface-barrier detector, is reverse biased at 80 volts, and the dead times of the ND575 Analog-to-Digital Converter (ADC) and ND66 Multi-Channel Analyzer are 60 n-seconds and

6 p-seconds, respectively. Therefore, no dead-time correction is necessary, as long as the counting rate is kept lower than 1,000 counts per second [16] (or the averaged time elapse between two counts is higher than 1,000 p-seconds).

A typical full-energy spectrum measured in these experiments is shown in Figure 6-2 in semi-logarithmic scale. The spectrum is shown on a display screen while accumulating counts and the final result, after a chosen time T, is stored in the memory units of the ND66. The Full Width Half Maximum (FWHF) of the spectrum can be found by moving the cursor on the display, which indicates the number of counts under each energy channel. The systematic range used in these experiments was from peak channel -FWHM x 6 to peak channel +FWHM x 2. Therefore, the total number of counts MT, which will be analyzed later, is always under a fixed portion of the full spectrum.












Ortec 210 Detector
Control
Unit
Detector Ortec 109


I.- - - Pre-Amp.

Vacuum
Chamber


SouIrce:
241


95 Am


Ortec 410

Multimode Amplifier




ND 575
Analog
to


Digital Convertor


ND 66
MultiChannel Analyzer


Block diagram counting system


&1 54


Figure 6-1.






































* t.. .- .~..


50 100


250


ENERGY CHANNEL


Figure 6-2.


Typical full-energy spectrum









The MT'S of adjacent time intervals can be read directly from the memory units of the ND66 Multi-Channel Analyzer; thus the Allan variance can be calculated by


A2_ 1 [ 1 N- I ) _ M.i+l)]2 (6-1)
aM T N 7J T T


since



1 M.i) (6-2)



the relative Allan variance, R(T), defined by Equation (5-15), can be found by using Equations (6-1) and (6-2).

Part of the data, mostly the total number of counts for T longer than 1,000 minutes, was not read directly from the ND66 Multi-Channel Analyzer. An "add-up" method was adopted; namely, MO) was obtained by
Analyzer. ~1000waobindy
adding up M() and M(2), and M(2)000 equals the sum of M(500 and M(4) etc. Physically, since M500s were measured in adjacent time intervals, of course the first two can be added up as the total number of counts for the first 1,000-minute interval, and the third and the fourth can be summed up as M(2) However, in order to check the validity of this
1000
method, the following experiment has. been done.

By making use of a T-connector, the output signal of the Ortec 410 amplifier was fed simultaneously into two ND575 ADCs (see Figure 6-1). The first one (ADC #1) counted 100-minute measuremenst for 310 times, and the second one (ADC #2) accumulated 500-minute counts for 62









times; hence, both ADCs covered exactly the same time span. The "addup" method was applied to Ml00s, obtained from ADC #1, to find out the calculated M500S.

Table 6-1 lists the results calculated from both ADCs for

500-minute measurements. The difference between them, in each category, is <1%. This shows the validity of the "add-up" method.


Table 6-1.


Comparison of the results obtained from the "add-up" method (ADC #1) and the real-time measurements (ADC #2)


ADC #1 ADC #2

9,062,464.36 9,067,009.84 0A2
aM5007,862,171.3 7,943,285.5 R(500) 9.573 x 10-8 9.662 x 10-8













CHAPTER VII
COUNTING RESULTS


7.1 Convergence of the Allan Variance for N

Because the existence of a "variance of the variance" (or, a noise of the noise), the relative Allan variance itself is a fluctuating parameter. In order to obtain an accurate value of R(T), a sufficient number of measurements must be made especially when T is short. To determine the minimum number of measurements needed, we plot R(T) versus N for different T's. These figures show that when N is small, R(T) is spread over a wide range. When T is increased the spread in R(T) diminishes and finally R(T) converts to a stable value. For example, Figure 7-1 shows that for T = 100 minutes, 21 measurements of M(1O0) yield an R(1O0) = (7.09 � 3.94) x 10-7; for N = 24, R(T) = (7.23 � 2.14) x 10-7; for N = 27, R(T) = (7.13 � 1.85) x 10-7. Here R(T) is given in the form of (mean value � standard deviation). From Figure 7-2 we know that for T = 1 minute at least 70 measurements are necessary for a reliable value of R(T). Figure 7-3 shows that for T = 3 minutes we need N > 50.

7.2 Results of the Relative Allan Variance versus l/T

From Equation (5-15) we have


R(T) = 1_ + 2C' n 2 (7-) moT




















40 1


30 F


20 I


10 I


Figure 7-1. R(T) vs. N for T = 100 min.




















6 5 In
0
x 4



3 2 1 0


Figure 7-2 R(T) vs. N vor T = 1 min.





















I

x 3



2
I-r


2�a





00

0 I I60
0 20 40 60

N


Figure 7-3. R(T) vs. N for T = 3 min.









where C' is a constant, m0 is the averaged counting rate, and T is the counting time.

Figures 7-4 to 7-10 show seven series of experimental results of R(T) versus I/T. Figures 7-4 to 7-6 show clearly that for T small (I/T large), R(T) is equal to the value of Poissonian shot noise, the first term of Equation 7-1. But for T large (l/T small), the values of R(T) is larger than the value for shot noise. Figures 7-7 to 7-10 show the results of some experiments which were concentrated on longtime measurements. They all show that for T long enough the experimental values of R(T) are higher than 1/m0T and the curves start to level off, which implies the existence of 1/f noise. The second, constant term in Equation (7-1) becomes dominant.


7.3 Results of the Allan Variance versus T

Figures 7-11 and 7-12 give the plot of the Allan variance, TA2 versus T corresponding to the measurements of Figures 7-4 and 7-5, respectively. Recall that for shot noiseMT = mT, which is proportional to T, but for 1/f noise cA2 = 2CT2 kn 2, which is proportional OMT
to T2.

Since these figures (Figures 7-11 and 7-12) are plotted on log-log scale, the slope of log (aA) versus log(T) is really the exponent of T in the expression of MT see Equation (5-14). Figures 7-11 and 7-12 show clearly the dependence of A2 on T2 and T while dMT
different noise terms dominate the noise spectrum.




67




















-Poisson Noise .104.0 Exp. Result




a








10-6
70



10







-3 -2 -1
10 10 10


T (min.) Figure 7-4. R ep(T) vs. 1/T for m = 27,700/min.























- Poisson Noise


-4 0 Exp. Result 10







100



x
: -6 10
1
0










I . . ... ... . . .....I . . ..
-3 -2 -1
10 10 10 1
! mi.-"


Figure 7-5. Rexp (T) vs. I/T for mo = 17,980/min.




























-4
10





-5
10



0.


-Poisson Noise (D Exp. Result


-3 -2
10 10


-i
10
(min'")


Figure 7-6. R exp(T) vs. 1/T for mo = 18,160/min.


I -


































-5
10


-6
10


-3 -2
10 0 10 1I


~1.(mm:")


Figure 7-7. R exp(T) vs. I/T for m� = 17,180/min.




71




















-Poisson Noise


) Exp. Result








0- 5
10.


I.
x
,v -6
10

0


-7
10






-3 -2 -1
10 10 10 1(..i) r mn'


Figure 7-8. R (T) vs. 1/T for m = 18,356/min.
exp o



































-5
10


-7
10


-Poisson Noise


0 Exp. Result


-2
10


Figure 7-9. R exp(T) vs. 1/T for mo = 17,807/min.


10 ( i)














































10






1-7






-8
10


Poisson Noise


0 Exp. Result


0
~ /


-3
10


-2
10


Figure 7-10. Rexp (T) vs. I/T for mo = 18,180/min.


T. (min.')




74
















109






108






10 0
w Sop=2


4


QCQ
Z 6
10




0

10

Slope 10 2 3 T (min.)

1 10 10 10








Figure 7-11. A (T) vs. T for m= 27,700/min.
M T




75
















10 9












7
10
z slopes Z 0

z
6
10 0


0


10





4
10
2 3 4 1 10 10 10 10
T (m in.)







Figure 7-12. aA2 (T) vs. T for mo = 17,960/min.
'"T













CHAPTER VIII
DISCUSSION OF RESULTS


The curves shown in Chapter VII are not smooth, which is due to the nature of the noise fluctuations. In order to obtain a universal curve, the following procedure has been carried out.

Assume the experimental results of R(T) contain a fluctuation term AR, due to the presence of "noise of noise," i.e.,


R exp(T) =- mo +2C' kn 2 + AR (8-1)



where the average value of AR, , should be zero. The experiment was then repeated for several times, and the average of R exp(T),


= --Lo + 2C' kn 2 + (8-2)
exp m 0T


is obtained. Although the value of in Equation (8-2) can hardly be exactly zero for a finite number of measurements, it should be reduced by a great deal compared with the value of AR for a single measurement. Now, the value of gives the best estimation to the true value of R(T).

The distance between the radioactive source and the detector was adjusted such that very close counting rates were obtained while repeating the measurements. However, it is difficult to obtain identical









counting rates. Therefore, the shot noise term in the relative Allan variance, R(T), is slightly different for each series of measurements.

Before the average of R (T) is taken, the shot noise term, exp
1
, should be normalized to the same counting rate. Here a rate of
0
18,000 counts per minute was chosen. The normalized relative Allan variance Rn(T) is then

Rn(T) = R (T)- 1+ 1 1 + 2C' kn 2 nmexp oT 18,000 x T 18,000 x T


(8-3)

Tables 8-1 and 8-2 show the values of R exp(T) and R n(T). Instead of , is now used to estimate the true value of R(T). The value of is calculated as follows:


ERni (T) x DF.
1DF (8-4) n EDF.
1


where R ni(T) is the value of R n(T) obtained from the ith series and DFi stands for the degrees of freedom of that particular value, which equals the number of measurements minus one.

Figure 8-1 shows the comparison of and (1/18,000 x T) + 1 x 10-7 versus l/T, which suggests that the value of 2C' Zn 2 is about 1 x 10-7.

Figure 8-2 shows the comparison of the average value of the normalized Allan variance, ,w







Table 8-1. The values of Rexp(T) and Rn(T) for T = 1-50, all values x 10-7, the numbers in parentheses
being the number of measurements


Time (min.)
m 1 2 5 10 20 50
0


27,700/min.


17,980/min. 18,160/min.


17,807/min.


R (T) exp R n(T)


R (T) exp R n(T)


R (T) exp Rn (T)

R (T)
exp Rn (T)


348.7 544.1(96) 572.9 572.5(177) 504.8 509.8(248)


205.3 303.0(48) 232.1 231.9(88) 263.1

265.6(124)


84.27 123.30(19)


149.6 149.6(93)


71.85 72.85(49)


39.07 58.58(38) 41.16 42,78(83)


66,03 66.55(88)


11.91 21.66(18) 22.68 23,49(41) 34.11 34.37(44)


4.957

8.844(7)


11.65 11.970(16) 11.350 11.450(149) 12.150 12.030(56)







Table 8-2. The values of Rexg(T) and Rn(T) for T
ses being the number of measurements


= 100-3,000, all values x 10-7, the numbers in parenthe-


Time (min.)
m 100 200 500 1,000 2,000 3,000


27,770/min.


17,980/min.


18,160/min.


17,180/min.


18,356/min.


R n(T) exp(T) R (T)
n




Rexp (T) R (T)


Rexp(T)

R (T)
n



R n(T) Rexp(T
R (T)
n


4.126

6.075(18)


6.359

6.348(72)


7.398

7.437(310)


7.192

6.927(26)


6.003

6.111(30)


3.281

4.255(9)


4.093

4.087(36)


3.505

3.524(155)


8.390

8.257(13)


1.970

2.024(15)


6.105

6.494(3)


3.404

3.401(14)


0.9662

0.9742(62)


8.9990

8.9460(5)


2.3430

2.3640(6)


6. 8300

6.8290(7)


0.8133

0.8175(31)


1.3830

1.3850(15)


3.360

2.3610(10)


4.4280

4.4390(3)








Table 8-2-Continued


Time (min.)
mo 100 200 500 1,000 2,000 5,000


R exp(T) 8.147 4.263 2.3630 11,807/min. ex
R n(T) 8.087(28) 4.233(14) 2.3510(5)

Rexp (T) 3.808 1.3200 0.8042 0.6299 0.5865 18,180/mi n.0
Rn(T) 3.836(40) 1.3310(30) 0.8099(15) 0.6327(7) 0.5884(5)






Page
Missing
or
Unavailable

























Theoretical Value


0 Exp. Result


Figure 8-2.


< 2T)> vs. T for mo
MT


18,000/min.


109





8
10





7
10

z
4
4

z
6 S106




0
z
105


T (min.)









= x (18,000 x T)2 (8-5)



Very good agreement is obtained as shown in this figure.

The value of and 1/18,000 x T (shot noise level) are
n
listed in Table 8-3. It shows clearly that at this counting rate, 18,000 counts per minute, for T longer than 100 minutes, 1/f noise becomes noticeable. For T longer than 1,000 minutes, 1/f noise totally dominates the noise spectrum.

Besides the fluctuation of scattering cross section described in Handel's theory, two other possibilities may contribute to the fluctuation in the total number of counts. First, the pulse height, which is produced when an a-particle is absorbed by the detector, is a strong function of the bias voltage applied to the detector [17]. If the bias voltage was unstable, then a-particles with the same energy would have been registered in different energy channels. Fortunately, the detector control unit used in these experiments, Ortec 210, has a very good stability: bias voltage variation with line voltage is <�0.005% for 105-125 V AC input, and thestability is �0.01%[18]. Secondly, since a fixed portion of the full spectrum is counted, whether particles with energies around boundaries fall in or out of boundaries contributes fluctuations to the total number of counts.

Therefore, a wider energy range should have less fluctuations

in the total number of counts in a certain time period. This is indeed the case; see Table 8-4. However, for T = 100 minutes, the difference between R(T)'s for wider range and narrower range is negligible. For

















Table 8-3.


The values of and 1/18,000xT, all values x 10-7


Time (min.) 1 2 5 10 20 50 537.4 261.0 123.3 55.66 27.86 11.56 1 555.6 277.8 111.1 55.56 27.78 11.11
18,000 x T




Time (min.) 100 200 500 1,000 2,000 3,000 7.156 3.825 1.799 1.6480 1.1590 1.8160

1
1 5.556 2.778 1.111 0.5556 0.2778 0.1852
18,000 x T








Table 8-4. Comparison of experimental results for narrower energy
range (peak channel-FWHM x 6 to peak channel + FWHM x 2)
and wider energy range (peak channel-FWHM x 12 to peak
channel + FWHM x 6)


Time (min.) Channels 127-147a Channels 112-157a


A2



R(100)






A2
11500


R(500)




A2
M3000

R(3000)


1,812,492.94 2,430,286.7

-7
7.398 x 10



9,067,009.8 7,943,285.5 9.662 x 10-8



54,398,712.1 698,293,362 2.360 x 10-7


apeak channel: 142; FWHM:


2.5 channels


500


1,828,630.89 2,472,416.52 7.394 x 107




9,147,489.02 7,762,255.74 9.277 x 10-8




54,881 ,014.1 634,854,514 2.108 x 10-7


3,000









T = 50 minutes, the difference is 4%, but the value of is
n
62% higher than the shot noise level (see Table 8-3). For T = 3,000 minutes, the difference is 11%, and the value of is ten times the value for the shot noise level. Hence the energy range chosen in these experiments affected little the final results.

In certain R(T) versus l/T figures in Chapter VII and in

Figure 8-1, when T becomes large R(T) shows the tendency to be proportional to T. This phenomenon is not exclusive, since according to the Allan variance theorem R(T) is indeed proportional toT for very slow Lorentzian flicker noise [15].












CHAPTER IX
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK


From the experimental data presented in the Appendix, we see that the ratio for the averaged value of the total number of counts, , to the variance of M is very close to one when T is less than 100 minutes, which is an indication for Poissonian statistics. However, when T is greater than 100 minutes this ratio becomes greater than unity and generally increases monotonically with T, which indicates super-Poisonian statistics. With the help of the newly devised Allan variance theorem, we found that this excess noise in the radioactive decay rate has a 1/f spectrum.

Quantitatively, from Equation (5-18), we know the flicker floor F has a value of


F = 126.4 x 10- x (9-1)


To the author's knowledge, nobody has ever measured the dielectric constant of AmO2, the ct-particle source used in these experiments; thus the value of K in Equation (9-1) is unknown. However, since the atomic structure of AmO2 is similar to that for U02, the dielectric constant of UO2 (20.4 � 1.5 [19], 21.7 � 0.5 [20], and 21.0 � 1 [21]) can then be used as a reference.

If one considers the following factors that (1) the coherence factor, , is always less than unity; (2) the dielectric constant of









AmO2 may be larger than 21, then the measured value of F = 1 x 10-7 is in the right ballpark to verify Handel's theory. In particular, for C = 0.17, K = 21, one obtains F = 1.02 x lO, in accord with the observed value of Figures 8-1 and 8-2. We believe, therefore, that these experiments constitute the first experimental verification of Handel's quantum 1/f noise theory.

The data presented here all referred to 95Am241. If an a-source with different energy peaks is used, the counting could then be done in each of these energy peaks. This should result in an R(T) versus l/T as given in Figure 9-1.

If Handel's theory is correct, the flicker floor should be

proportional to v /c , i.e., to the energy E. Thus, the flicker floors shown in Figure 9-1 should have the ratios of these energy peaks. This experiment will therefore be a very strong test to verify the applicability and correctness of Handel's theory.






































-6
10


/



E1 > E2 > E3


1o-3 10-2 I0-1


Figure 9-1. R(T) vs. 1/T for different energy peaks






























APPENDIX
EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS







Experimental data for m0 = 27,700


var A2
T N var. V MT Rexp (T)


27,761 55,525 138,797 277,369 554,689

1,386,606 2,772,026 5,544,053

13,855,521


29,759 64,254 174,979 271,146 407,757

1,556,691 13,322,534 49,414,157 202,833,876


1.07

1.16 1.26

0.98 0.74 0.83

4.81 8.91

14.64


26,874 63,300 162,334 300,625 366,675 950,741

3,170,123 10,084,511 117,194,752


1

2

5

10

20 50 100

200 500


3.487

2.053 8.427

3.907 1.191

4.945 4.126 3.281 6.105


x lo-5 x 10-5 x lo-6 x lo-6 x 10-6 x lo-7 x 10-7 x lo-7 x 10-7


Table A-1.








Table A-2. Experimental data for m0 = 17,980


TNvr var A29 TQq N N>vr R (T) . MT exp


17,987 35,970 89,984 185,379 370,752

926,836 1,796,155 3,592,309

8,980,092 17,960,183


17,155 29,055 108,003 154,777 324,936 868,989

15,133,266 55,659,746 324,379,609 1,311,457,070


0.95 0.81 1.20 0.84 0.88

0.94 8.43

15.49 36.12 73.02


18,534 30,030

121,148 141,464 311,757 1,000,977 2,051,505 5,282,260 27,448,241 220,310,187


1

2 5 10

20 50 100

200 500

1,000


5.729 2.321 1.496 4.116

2.268 1.165

6.359 4.093

3.404 6.830


x 10-5 x 10-5 x 10-5 x lo-6 x 1o-6


x lo-6 x lo-7 x 10-7 x lo-7 x 10-7








Table A-3. Experimental data for m0 = 18,160


yar. A2 R (T) T N var y OMT exp


18,160 36,321

90,795 181,684 363,367 908,647 1,812,493 3,625,004

9,067,010 18,134,020 36,265,809 54,398,714


16,829 34,706 64,814 214,732 436,054 1,274,359 9,116,358 31,272,439

168,503,103 668,501,571 2,626,291 ,950 5,991,195,130


0.93 0.96 0.71 1.18 1.20 1.40

5.03 8.63 16.58 36.86 72.42 110.14


16,647

34,714 59,228 217,969 450,422 936,880

2,430,287 4,606,285 7,943,286 26,745,242 181,946,661 698,293,362


5.048 x 10-5 2.631 x 10-5

7.185 x 10-6 6.603 x 10-6 3.411 x 10-6 1.135 x 10-6 7.398 x 10-7 3.505 x 10-7 9.662 x 10-8 8.133 x 10-8 1.383 x 10-7 2.360 x 10-7


1 2 5 10

20 50 100

200 500 1,000

2,000 3,000













Table A-4. Experimental data for m 0 17,180


var A2
>var < TRMT R (T)



100 26 1,717,908 10,533,131 6.13 2,122,404 7.192 x 10-7 200 13 3,435,815 38,770,910 11.28 9,904,640 8.390 x 10-7

-7
500 58,589,200 186,528,449 21.72 66,392,064 8.999 xI0




Full Text
Table 8-4. Comparison of experimental results for narrower energy
range (peak channelFWHM x 6 to peak channel + FWHM x 2)
and wider energy range (peak channelFWHM x 12 to peak
channel + FWHM x 6)
Time (min.)
Channels 127-147a
Channels 112-1573
o
o
V
1,812,492.94
1,828,630.89
100
A2
M100
2,430,286.7
2,472,416.52
R(100)
7.398 x 107
7.394 x 10'7
A
o
o
LD
5
9,067,009.8
9,147,489.02
500
A2
G|V|500
7,943,285.5
7,762,255.74
R(500)
9.662 x 10"8
9.277 x 10"8
A
o
o
oo
5
54,398,712.1
54,881,014.1
3,000
A2
aM
"3000
698,293,362
634,854,514
R(3000)
2.360 x IQ"7
2.108 x 10-7
aPeak channel: 142; FWHM: 2.5 channels


15
Figure 2-3.
Var X + M2
vs. M for k = 0.016 in log-log scale


velocity change in the scattering process. The relative scattering rate
2
density with energy loss e, |b(e)| is obtained by dividing w(f) by the
energy of a photon e = hf:
|b(f)|2 = 4efiAVli = ml-fL (5-4)
3c32,rtrf,c Tc 37rfK fK
|b(S)l2 = 4e£(AV)i = CCA (5-5)
3c3he< eK
where
A = 2(AB)2 .w = AV
A 3tt l c
e2
and a = is the fine structure constant. In the MKS system, a =
Tvc
where Kq is the dielectric constant of vacuum. The spectral den
sity of the relative fluctuations is
(5-6)
TTckq
S ^ = 2[1 + aA An (A/e)] ^aA/fic
kl
2aA/fx
(5-7)
5.2 The Allan Variance Theorem
The main link between counting statistics and particle current
noise is provided by MacDonald's theorem,
d xahA 1
7T
r
S (co)oj ^sin uTdc
, m
(5-8)


89
T
Figure 9-1. R(T) vs. 1/T for different energy peaks


Table 2-
^ ypole
(N) and y
(N) for
different
values of k, all values
X
o
1
CT>
N
k
1
2
3
4
5 6
7
8
9
10
Ppole
0.02
141,500
57,960
35,050
24,820
19,130 15,520
13,040
11,240
9,868
8,793
0.016
126,500
50,720
30,360
21,370
16,400 13,270
11,130
9,575
8,398
7,476
0.014
118,322
46,807
27,833
19,520
14,944 12,073
10,112
8,692
7,618
6,778
y
Hmax
0.02
23,580
16,560
13,140
11,030
9,565 8,465
7,607
6,917
6,343
5,862
0.016
21,080
14,490
11,390
9,498
8,200 7,238
6,493
5,892
5,399
4,984
0.014
19,720
13,373
10,437
8,676
7,472 6,585
5,899
5,349
4,897
4,519


var X +M
18
Figure 2-6.
Var X + M
2
vs.
M for k = 0.016 in linear scale


Figure 3-2. Block diagram of the noise measurement system


73
Figure 7-10. RgXp(T) vs. 1/T for mo 18,180/min.


CHAPTER VIII
DISCUSSION OF RESULTS
The curves shown in Chapter VII are not smooth, which is due
to the nature of the noise fluctuations. In order to obtain a univer
sal curve, the following procedure has been carried out.
Assume the experimental results of R(T) contain a fluctuation
term AR, due to the presence of "noise of noise," i.e.,
Rexp(T) = iTT+ 2C £n 2 + AR (8-1)
^ o
where the average value of AR, , should be zero. The experiment was
then repeated for several times, and the average of ReXp(T),
= HTT + 2C' £n 2 + (8-2)
is obtained. Although the value of in Equation (8-2) can hardly be
exactly zero for a finite number of measurements, it should be reduced
by a great deal compared with the value of AR for a single measurement.
Now, the value of gives the best estimation to the true value
of R(T).
The distance between the radioactive source and the detector was
adjusted such that very close counting rates were obtained while
repeating the measurements. However, it is difficult to obtain identical
76


Table 8-3.
The values
of (T)> and
1/18,000 xT,
all values
x 10"7
Time (min.)
1
2
5
10
20
50
537.4
261.0
123.3
55.66
27.86
11.56
1
18,000 x T
555.6
277.8
111.1
55.56
27.78
11.11
Time (min.)
100
200
500
1 ,000
2,000
3,000
7.156
3.825
1.799
1.6480
1.1590
1.8160
1
18,000 x T
5.556
2.778
1.111
0.5556
0.2778
0.1852


BIOGRAPHICAL SKETCH
Jeng Gong was born in Taipei, Republic of China, on July 18,
1953. He attended the National Cheng Kung University in Tainan,
Taiwan, and received the Bachelor of Science degree in electrical
engineering in 1975. After entering the Chinese Army from 1975 to 1977,
he worked with Dah-Shen Electronic Company, Taipei, until 1978.
Entering the University of Florida in 1979, he received the
Master of Engineering degree in electrical engineering in 1980. From
1981 to the present, he has pursued his work toward the degree of
Doctor of Philosophy.
In 1978, he married the former Chang-Ling Tseo and is the
father of one son, Dow.
100


PART A
NOISE FROM HOLE-INITIATED
PHOTO AVALANCHE PROCESSES


Table 8-1.
The values
being the i
of Rexp(T) anc* Rn(T) fr T =
number of measurements
1-50, all values x 10 the
numbers in
parentheses
Time (min.)
m
0
1
2
5
10
20
50
27,700/min.
Rexp
Rn
348.7
544.1(96)
205.3
303.0(48)
84.27
123.30(19)
39.07
58.58(38)
11.91
21.66(18)
4.957
8.844(7)
17,980/min.
WT>
Rn
572.9
572.5(177)
232.1
231.9(88)
149.6
149.6(93)
41.16
42.78(83)
22.68
23,49(41)
11.65
11.970(16)
18,160/min.
Rexp
Rn
504.8
509.8(248)
263.1
265.6(124)
71.85
72.85(49)
66,03
66.55(88)
34.11
34.37(44)
11.350
11.450(149)
17,807/min.
Rexp
Rn'T>
12.150
12.030(56)


2
Whereas most of the noise phenomena, like shot noise, thermal
noise, and generation-recombination noise, are well understood, 1/f
noise remains an enigma. This noise has been observed almost every
where: semiconductor devices, music, traffic flow, hourglass flow,
the frequency of sunspots, the light output of quasars, etc. Because
of its universality, some investigators believe that there must be
some universal phenomenon operative in all these manifestations [10-
12].
One of the theories of 1/f noise is the quantum theory, based
on the infrared divergence phenomenon and developed by Handel [10].
This theory is fundamental in the sense that it derives the 1/f
spectrum from basic quantum physics at the level of a single charged
particle subject to scattering, although the final result depends
essentially on the presence of many carriers, making 1/f noise similar
to diffraction patterns which are one-particle effects but can be seen
only if many particles are diffracted. In addition, the theory is
universal in the sense that any infraquanta with infrared-divergent
coupling to the current carriers will give a contribution to the
observed 1/f noise proportional to their coupling constant. Such
infraquanta are, for example, very low frequency photons, various types
of phonons, shallow electron-hole pairs on the Fermi surface of a
metal, spin waves, correlated states [11], etc.
Recently, this theory has been reformulated with quantum-
optical terminology and compound-Poisson statistics in a paper written
by Van Vliet, van der Ziel, and Handel, which led to the idea of


a 5.5 potentiometer give the correct bias voltage needed. The
diode current 1^ is monitored by a Keith!ey 602 solid-state elec
trometer which is capable of measuring current accurately to less than
1 picoampere. An N-channel FET source follower is set in between the
biasing circuit and the low-noise amplifier (LNA) to match the
impedances.
The 5722 vacuum noise diode is chosen as the standard noise
source because its known level of Gaussian white noise and proven per
formance. The complete circuit is shown in Figure 3-4. The anode
current is adjusted by varying the heater voltage, V^, with a car
battery to supply heater current. The anode load is a 100 millihenry
high Q miniature inductor.
The low-noise amplifier as designed by L. M. Rucker of the
University of Florida is shown in Figure 3-5; it uses a high trans
conductance JFET, type 2N6451 as the input of a cascade input stage.
The output is a bipolar transistor, type 2N3904. The input stage has
an overall gain of 30 dB. The second and third stages are identical
20 dB gain cascades using 2N4220A FET inputs and 2N3904 bipolar outputs.
Because the gain of the first stage, no special effect was made to
reduce the noise in the later stages. The final stage is a unity gain
buffer having an output impedance of approximately 50 ohms. The
buffer is connected to one of the earlier stages through a rotary
switch having a low capacitance between the contacts. The overall gain
of the amplifier may then be set to 30, 50, or 70 dB. The bandwidth
depends on the gain setting and is 3.5, 2.5, and 2.2 MHz, respectively.


86
T = 50 minutes, the difference is 4%, but the value of is
n
62% higher than the shot noise level (see Table 8-3). For T = 3,000
minutes, the difference is 11%, and the value of is ten
times the value for the shot noise level. Hence the energy range
chosen in these experiments affected little the final results.
In certain R(T) versus 1/T figures in Chapter VII and in
Figure 8-1, when T becomes large R(T) shows the tendency to be propor
tional to T. This phenomenon is not exclusive, since according to the
Allan variance theorem R(T) is indeed proportional to T for very slow
Lorentzian flicker noise [15].


verifying the theory on a "clean" system outside the domain of solid-
state physicsradioactive a-decay [13].
The purpose of this study is to examine the existence of 1/f
241
noise in radioactive decay from g^Am a-particle source. The results
are presented in Part B of this dissertation.


APPENDIX
EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS


vacuum tube diode, I
Now,
A*
M2 GB
(3-2)
where G, B, and are the same as in the previous case, S^-j = 2ql^
is the spectral density of the calibration noise source where q is
electronic charge.
Finally, we turn on the light and turn off the calibration
noise source and obtain a new reading M^,
Mo = GB(S. i + ST + S-, )
3 v bkg I, laser'
(3-3)
where is the noise spectral density due to avalanche processes, and
St is the noise introduced by the laser light source which is con-
laser J 3
sidered as constant background noise. Here we assume Sjd and S^gser
are independent.
The equivalent saturated diode current for + S-]aser, I ,
can then be obtained by calculating
I
eq
- M'
- M
lA
(3-4)
The procedure is repeated for various bias voltages of the
DUT. Experimental results are plotted and compared with the theory given
in Chapter II.


NUMBER OF COUNTS
Figure 6-2. Typical full-energy spectrum


bb
According to Handel's theory, the constant C in the 1/f noise
term in the relative Allan variance is
C = 2oA£/k (5-16)
where 5 is a coherence factor; for a-particles it is expected to be close
2 2
to one; a is the fine structure constant 1/137, and A = 2(Av) /3ttc
where Av is the velocity change of the particles in the emission process,
c is the velocity of light. For a-particles one finds
= 8.32 x 10"7 x -L (5-17)
< K
where E is energy in MeV and k is the dielectric constant of the radio
active material.
In these experiments E = 5.48 MeV, so that the value of 2C £n 2
(the value of the flicker floor, F) is
F = 4? 8~32 x 7 x 5-4-8- x jn 2 126.4 x 107 x -£-
K K
K
(5-18)


Page
Missing
or
Unavailable


NOISE ASSOCIATED WITH ELECTRON STATISTICS
IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF a-PARTICLES
By
JENG GONG
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

ACKNOWLEDGMENTS
I am deeply indebted to Dr. Carolyn Van Vliet, Dr. Aldert
van der Ziel, and Dr. Peter Handel for their most generous and valuable
guidance and assistance during the preparation of this work. I thank
Dr. Alan Sutherland, Dr. Eugene Chenette, and Dr. Gys Bosman for their
advice and encouragement. Dr. Wiliam Ellis has been most helpful
during many phases of this work and his help, especially in discussions
of radiation detection techniques, is greatly appreciated.
I also wish to express my appreciation to Dr. R. J. McIntyre
and Dr. Dean Schoenfeld for providing RCA diodes and the Argon-ion
Laser, respectively. I am particularly indebted to Mrs. S. L. Wang for
the preparation of the figures presented in this work.
My deepest gratitude goes to my parents, my wife, and my son
who have always had faith, encouragement, and understanding when it
was most needed.
This research was supported by AFOSR contract, #82-0226.
11

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
ABSTRACT v
CHAPTER I: INTRODUCTION 1
PART A
NOISE FROM HOLE-INITIATED
PHOTO AVALANCHE PROCESSES
CHAPTER II: THEORY OF NOISE IN AVALANCHE DIODES 5
CHAPTER III: DEVICE DESCRIPTION AND EXPERIMENTAL SET-UP
FOR MEASURING PHOTODIODE NOISE 20
CHAPTER IV: RESULTS FOR HOLE-INITIATED AVALANCHE CURRENTS 29
PART B
EMISSION STATISTICS OF ct-PARTICLES
CHAPTER V: THEORETICAL PERSPECTIVES 49
5.1 Handel's Theory 49
5.2 The Allan Variance Theorem 52
5.3 Application of the Allan Variance
Theorem to Counting Statistics 54
CHAPTER VI: EXPERIMENTAL SET-UP FOR LONG-TIME COUNTING 57
CHAPTER VII: COUNTING RESULTS 62
7.1 Convergence of the Allan Variance
for N-> 62
7.2 Results of the Relative Allan
Variance vs. 1/T 62
7.3 Results of the Allan Variance vs. T .... 66
CHAPTER VIII: DISCUSSION OF RESULTS 76
i i i

Page
CHAPTER IX: CONCLUSIONS AND RECOMMENDATIONS FOR
FUTURE WORK 87
APPENDIX: EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS .... 91
REFERENCES 98
BIOGRAPHICAL SKETCH TOO
IV

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOISE ASSOCIATED WITH ELECTRON STATISTICS
IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF a-PARTICLES
By
Jeng Gong
August 1983
Chairperson: C. M. Van Vliet
Major Department: Electrical Engineering
It is known that the standard theories of avalanche statistics
for two-carrier impact ionization by Tager, McIntyre, and Personick
only deal with processes for which the number of possible ionizations
is very large. On the contrary, it is believed that in many modern
devices the number N of possible ionizations is finite and perhaps even
very smal1 (N = 1 5).
In 1978, van Vliet and Rucker developed a new theory for-this
small N case, which has been fully confirmed for the electron-initiated
ionization process. Part A of this dissertation presents the results
of a detailed noise study with hole-initiated avalanche currents which
corroborate van Vliet and Rucker's theory for the second time.
Data obtained from extensive measurements on counting techniques
241
for a-particles radioactive decay from g^Am are presented in Part B
v

of this dissertation. These data have shown that the statistics are
non-Poissonian for large counting times (order 1,000 minutes) in
contrast with the fact that many textbooks cite a-decay as an example
for Poisson statistics.
Detailed measurements of the Allan variance indicated a
"flicker floor" due to the presence of 1/f noise in the decay, of
10"7. This result is in agreement with Handel's quantum 1/f noise
theory. If upheld by further measurements, then this would be the
first quantitative indication that 1/f noise is caused by emission of
long wavelength infraquanta, such as soft photons causing minute
inelastic losses in the scattered wave packet.
vi

CHAPTER I
INTRODUCTION
Avalanche multiplication in devices occurs when the free
carriers gain enough energy in an electric field so that they can
ionize bound carriers upon impact. This phenomenon has been observed
in a number of devices, such as reverse biased p-n junctions, impact
diodes, Read diodes, FET's, etc. Most authors consider the ionization
process as a continuous process [1-6], which means that either the
region over which the avalanching occurs is very long, or the number
of ionizing collisions per primary carrier transit N is infinite or at
least very large.
Ionizations can occur due to electron or hole impact. The
primary carriers responsible for the ionization can be either thermally
generated, or arise from tunneling (as in Zener diodes), or stem from
light absorption; this is the case in avalanche photodiodes, as con
sidered in these studies.
For present-day small dimension devices, the number of ioniza
tions per carrier transit N is small. In 1978 van VIiet and Rucker
used a new statistical approach, referred to as the "method of recurrent
generation functions," to develop a new theory which is valid for an
arbitrary number of N [7,8]. This theory has been corroborated for
electron-initiated ionization process [9].
Measurements with hole-initiated avalanche current are made and
investigated; the resul ts wi 11 be discussed in Part A of this dissertation.
1

2
Whereas most of the noise phenomena, like shot noise, thermal
noise, and generation-recombination noise, are well understood, 1/f
noise remains an enigma. This noise has been observed almost every
where: semiconductor devices, music, traffic flow, hourglass flow,
the frequency of sunspots, the light output of quasars, etc. Because
of its universality, some investigators believe that there must be
some universal phenomenon operative in all these manifestations [10-
12].
One of the theories of 1/f noise is the quantum theory, based
on the infrared divergence phenomenon and developed by Handel [10].
This theory is fundamental in the sense that it derives the 1/f
spectrum from basic quantum physics at the level of a single charged
particle subject to scattering, although the final result depends
essentially on the presence of many carriers, making 1/f noise similar
to diffraction patterns which are one-particle effects but can be seen
only if many particles are diffracted. In addition, the theory is
universal in the sense that any infraquanta with infrared-divergent
coupling to the current carriers will give a contribution to the
observed 1/f noise proportional to their coupling constant. Such
infraquanta are, for example, very low frequency photons, various types
of phonons, shallow electron-hole pairs on the Fermi surface of a
metal, spin waves, correlated states [11], etc.
Recently, this theory has been reformulated with quantum-
optical terminology and compound-Poisson statistics in a paper written
by Van Vliet, van der Ziel, and Handel, which led to the idea of

verifying the theory on a "clean" system outside the domain of solid-
state physicsradioactive a-decay [13].
The purpose of this study is to examine the existence of 1/f
241
noise in radioactive decay from g^Am a-particle source. The results
are presented in Part B of this dissertation.

PART A
NOISE FROM HOLE-INITIATED
PHOTO AVALANCHE PROCESSES

CHAPTER II
THEORY OF NOISE IN AVALANCHE DIODES
The first theories for noise in avalanche diodes were given
by Tager (1965) [1] and by McIntyre (1966) [2]. Tager assumed that
the ionization coefficients for ionization by electrons and holes are
equal. Under these conditions the noise spectral density is given by
St = 2ql M3 (2-1)
ld p
where 1^ is the avalanche current, I is the primary photocurrent,
q is the electronic charge, M is the average gain, and where it is
assumed that the incoming particles show full shot noise.
The theory by McIntyre allowed for the possibility that the
ionization coefficients are unequal. Let a(x)dx be the probability
for ionization by an electron in the interval (x,x + dx) and let g(x)
be the ionization probability by a hole. For simplicity, McIntyre
assumed B(x) = ka(x). The noise spectral density is then given by
S
2a I M'
pr
1 (1 k)
M l'
2-t
M
/
(2-2)
for the case that the primary carriers injected into the avalanche
region are electrons; in case these carriers are holes, replace k by
1/k; if both occur, add the two partial results.
5

It is not necessary to commit oneself to Poisson statistics
of the primary particles; so, one can consider immediately the
statistics of the avalanche process. Thus let X denote the offspring
(plus original carrier) due to one incoming primary carrier. Then,
according to the variance theorem,
Sj = M2Si + 2qlpr var X (2-3)
where M = . From this we find that (2-1) and (2-2) are tantamount
to
var X = M^(M 1) (Tager)
(2-4)
and
var X = M(M 1) + k(M 1)^M (McIntyre) (2-5)
The above theories assumed that the region over which avalanche
occurs is very long, so that the number N of possible ionizations per
carrier transmit is very large. Therefore, these theories can be
referred to as asymptotic theories. Lukaszek et al. at the University of
Florida (1974) were the first authors to show that this assumption is
unrealistic [14], in particular for the region of onset (low gain) of
avalanche multiplication. They developed a theory for the case that N = 1
and N = 2. Besides for short avalanche regions, these formulas should
always be applicable at the onset of avalanche ionization. For N = 1,
their results give

7
var X = M(M 1) 1 +
IT 1%rrl-pj (Lukaszek) (2-6)
where X is the a priori chance for ionization by an electron after the
electron has gained enough energy to ionize, whereas y pertains to ioni
zation by a hole.
In 1978 van VIiet and Rucker reinvestigated the problem. A new
theoretical method was developed, named the "method of recurrent
generating functions" [7,8], By this method they were able to solve the
complete problem, in which N possible ionizations per carrier transit,
or per traveling hole-electron pair (the electron and hole going in
opposite directions) are possible. In this case the basic parameters
are the a priori ionization probabilities by electron impact (x) or by
hole impact (y), once these carriers have covered a path which is long
enough to gain the necessary ionization energy from the electrical
field.
For the case that the primary carriers injected into
the avalanche region are electrons, the following results are found
(1 + A)N(1 k)
(2-7)
(1 + kX)N+1 k(l + X)N+1
var X e <(AX)2> =
2
2 + X + kX
X
(2-8)

8
The asymptotic limit for N -> is not trivial; in the paper of van
Vliet and Rucker it is shown that (2-7) and (2-8) lead to McIntyre's
results.
For hole initiated avalanche case, replace k by 1/k, X by y,
and y by X. The results are
Mt¡ E fc =
(1
kN(l
k)[k(l
+ u)T
+ ur (k + y)
N+T
(2-9)
var
<(axJ¡)2> =
- DO k)
2k + yk + y
k-u2
k + y
x
" 1
1 + V
M
h 1
juni
- kj|
(2-10)
We now consider Equations (2-9) and (2-10) in more detail. At
the onset of ionization the field is just high enough to sustain one
possible ionization in the avalanche region; thus N = 1. With
increasing field, the value of y, denoted as y(l), increases and so
the gain increases according to Equation (2-9). The length £-j neces
sary to gain the ionizing energy simultaneously decreases, until two
ionizations per carrier transit are possible. The value of y(l),
just prior to this is denoted as uO)max- When two ionizations per
carrier transit are possible, the diode switches over from the regime
N = 1 to the regime N = 2. To realize the gain M of that operating
point, the value of y for the new regime, denoted as y(2), is consider
ably less than the value of y(l)m=v prior to the switch-over, as is
found by inverting (2-9) for y with M fixed, taking N = 1 and N = 2,

9
respectively. See Figure 2-1. At the M value for which the switch
over occurs, the variance jumps from var X-j to var 1^- Equation (2-10)
indicates that the decrease in y and the increase in N cause var X
to make a positive jump at this particular M, which we call a break
point value.
If now the field is further increased, the paths £-j +
decreases, y(2) increases, and M increases, until at y = y(2) three
TlaX
ionizations per carrier transit are possible. Once the regime N = 3 is
possible, the y(3) for these processes is much smaller than y(2) in
max
order to realize the same M. The switch-over marks another breakpoint
value for M, at which the variance jumps due to the reduction of y, from
y(2)max to y(3), etc. We thus obtain an overall curve of var X versus
M, which shows discontinuities at the breakpoint values of M, for which
the regime switches N -> N + 1. In order to construct this overall
curve we need only plot Equation (2-10) with y expressed as y(M,N)
according to the inversion of (2-9), where N = 1 and y = y(l) up to
y(1 )max next N = 2 and Ia = v(2) up to u(2)max> etc.
We conjecture that y(N)max is related to the pole value,
y (N), for which the denominator of (2-9) is zero. These pole values
r
are listed in Table 2-1. On general ground we assumed that the ymgx is
related to yp by
y(N)
N
max N + 5 Mp
yn(N)
(2-11)
The reason for this presumed relationship is that for very large
N we must have y(N) --y (N) in order that M can rise without saturation.
K max 'p

=0.016
Figure 2-1.
y vs. M

Table 2-
^ ypole
(N) and y
(N) for
different
values of k, all values
X
o
1
CT>
N
k
1
2
3
4
5 6
7
8
9
10
Ppole
0.02
141,500
57,960
35,050
24,820
19,130 15,520
13,040
11,240
9,868
8,793
0.016
126,500
50,720
30,360
21,370
16,400 13,270
11,130
9,575
8,398
7,476
0.014
118,322
46,807
27,833
19,520
14,944 12,073
10,112
8,692
7,618
6,778
y
Hmax
0.02
23,580
16,560
13,140
11,030
9,565 8,465
7,607
6,917
6,343
5,862
0.016
21,080
14,490
11,390
9,498
8,200 7,238
6,493
5,892
5,399
4,984
0.014
19,720
13,373
10,437
8,676
7,472 6,585
5,899
5,349
4,897
4,519

12
Converting Equation (2-3) to terms of equivalent saturated
diode current, we obtain
ST = M2St + 2ql var X = 2ql (2-12)
!d Ipr Pr ePd
When the primary current shows full shot noise, we have
Ian /I = M2 + var X (2-13)
eqd/ pr
while
W M f2-14)
Thus, suppose we plot theoretically log (M + var X) versus log (M).
Such a curve is identical to a plot of log (Ieqd) versus log (Id); the
origins of the two curves are displaced by a line under 45, the X-axis
and Y-axis displacements being log (I ). Therefore, by comparing the
experimental curve of Ieq^ versus Id on log-log paper with the theoreti-
2
cal curves for M + var X versus M on log-log paper, we find immediately
Ipr from the displacement of the origin. A special favorable feature
2
is that the theoretical curves of M + var X on log-log paper are very
close for different values of k. Hence k needs only be approximately
known for the determination of I Once I has been found, we then
pr pr
plot Ieq/Ipr vs. Id/Ipr on linear paper, and compare it with curves for
2
M + var X versus M on linear paper. From the magnitude as well as
from the breakpoints in the curve, k is obtained for the best fit. The

13
2
computer plots for M + var X versus M are obtained from (2-9) and
(2-10). The results are given in log-log form in Figures 2-2 to 2-4
and on linear paper in Figures 2-5 to 2-7.

14
Figure 2-2.
Var X + M
2
for k = 0.014 in log-log scale

15
Figure 2-3.
Var X + M2
vs. M for k = 0.016 in log-log scale

16
Figure 2-4.
Var X + M2
vs. M for k = 0.02 in log-log scale

17
Figure 2-5. Var X + M2 vs. M for k = 0.014 in linear scale

var X +M
18
Figure 2-6.
Var X + M
2
vs.
M for k = 0.016 in linear scale

va r X +M'
19
k : 0.0 2
Figure 2-7.
Var X + M vs,
M for k = 0.02 in linear scale

CHAPTER III
DEVICE DESCRIPTION AND EXPERIMENTAL SETUP
FOR MEASURING PHOTO DIODE NOISE
Two specially designed RCA "reach-through" diodes are used
in these hole-initiated avalanche process studies. Figure 3-1 shows the
diode structure and impurity concentration profile. This structure is
designed such that the depletion region in the p-type material reaches
through to the lightly doped it-region when the reverse-bias voltage is
about 200 volts. Further increase in the reverse-bias voltage quickly
depletes the tt-region.
A window is opened on the n+p junction side of the diode so that
light can be shone on the n+ region and then generate electron-hole
pairs. Because the diode is reverse biased, holes will drift into the
high-field region and initiate avalanche processes. The wavelength of
the light is chosen at 502.8 nm so that the absorption depth is shallow;
no light can penetrate to the p-type region, which excludes the possi
bility of electron-initiated avalanche multiplication.
The block diagram of the whole system is shown in Figure 3-2.
The light source is a Spectra Physics 171-08 Argon-ion Laser. The
TEM-01 mode is used. This laser gives a doughnut-shaped light which is
focused by a plain-convex lens. Figure 3-3 shows the biasing circuit
of the device under test (DUT). High Q capacitors are used, so as to
reduce the thermal noise as much as possible. Six 70-volt batteries and
20

3.4X1015 -*2/4Ln LD=1.7Hm
A '
n =
^1/2 I
TT L,
P =
5 X 1 012
TT 1 7 2 L
TT La
-V1
*"A5 7.6 pr
p"^ = 1 m deep
p m
Figure 3-1.
Diode structure and impurity concentration profile

Figure 3-2. Block diagram of the noise measurement system

5 pF-200 pF
70V-
70V-
70 V-
70 V-
70 V-
70V-
Figure 3-3. Biasing circuit of the DUT
h-KkKhl
20 V
O.I/iF 0.1/i.F 100ft J
7mH c

5.5M)
6)
(oj
NOISE
OUTPUT
>AV7
30M1
lV\AA/
IK
O.I/xF
2N 4220A
TO
LNA
IV
o

a 5.5 potentiometer give the correct bias voltage needed. The
diode current 1^ is monitored by a Keith!ey 602 solid-state elec
trometer which is capable of measuring current accurately to less than
1 picoampere. An N-channel FET source follower is set in between the
biasing circuit and the low-noise amplifier (LNA) to match the
impedances.
The 5722 vacuum noise diode is chosen as the standard noise
source because its known level of Gaussian white noise and proven per
formance. The complete circuit is shown in Figure 3-4. The anode
current is adjusted by varying the heater voltage, V^, with a car
battery to supply heater current. The anode load is a 100 millihenry
high Q miniature inductor.
The low-noise amplifier as designed by L. M. Rucker of the
University of Florida is shown in Figure 3-5; it uses a high trans
conductance JFET, type 2N6451 as the input of a cascade input stage.
The output is a bipolar transistor, type 2N3904. The input stage has
an overall gain of 30 dB. The second and third stages are identical
20 dB gain cascades using 2N4220A FET inputs and 2N3904 bipolar outputs.
Because the gain of the first stage, no special effect was made to
reduce the noise in the later stages. The final stage is a unity gain
buffer having an output impedance of approximately 50 ohms. The
buffer is connected to one of the earlier stages through a rotary
switch having a low capacitance between the contacts. The overall gain
of the amplifier may then be set to 30, 50, or 70 dB. The bandwidth
depends on the gain setting and is 3.5, 2.5, and 2.2 MHz, respectively.

5 722
No ¡se Diode
Noise Output
Figure 3-4. Circuit of a noise standard using a vacuum diode

NPUT
+20V
Figure 3-5. Circuit for a low-noise amplifier (LNA)

Immediately following the LNA is the HP-31OA Wave Analyzer,
which is used essentially as a narrow-band, high-gain, tunable filter.
It is used in the AM mode with a bandwidth of 3 KHz.
The output of the wave analyzer is connected, through an RC
low-pass filter (or an integrator), to the Y-axis input of the MFE 815
X-Y plotter with an MFE 7T time base on the plotter's X-axis input
terminals. The Y-direction displacement is proportional to the output
voltage of the HP-310 Wave Analyzer and its time average over a certain
period of time can be obtained from this plotter. This time-averaged
voltage reading gives more accurate results than that obtained directly
from an rms volt meter.
The whole system is first operated with both the light and
calibration noise source off. The variable capacitor (see Figure 3-3)
is tuned at the minimum capacitance position and the receiver (HP-310
Wave Analyzer) is set to the resonant frequency of the biasing circuit,
203 KHz, such that the maximum noise output is obtained. A voltage
meter reading M-j is recorded and the noise power at the output of the
aave analyzer in this case is
M
2
1
(3-1)
where B is the bandwidth of the whole system, G the power gain of the
amplifier, and stands for the spectral density of background
noise, which includes the amplifiers noise and noise due to the dark
current of the diode.
Next, we turn the calibration noise source on and record the
meter reading M^ and the average anode current of the 5722 saturated

vacuum tube diode, I
Now,
A*
M2 GB
(3-2)
where G, B, and are the same as in the previous case, S^-j = 2ql^
is the spectral density of the calibration noise source where q is
electronic charge.
Finally, we turn on the light and turn off the calibration
noise source and obtain a new reading M^,
Mo = GB(S. i + ST + S-, )
3 v bkg I, laser'
(3-3)
where is the noise spectral density due to avalanche processes, and
St is the noise introduced by the laser light source which is con-
laser J 3
sidered as constant background noise. Here we assume Sjd and S^gser
are independent.
The equivalent saturated diode current for + S-]aser, I ,
can then be obtained by calculating
I
eq
- M'
- M
lA
(3-4)
The procedure is repeated for various bias voltages of the
DUT. Experimental results are plotted and compared with the theory given
in Chapter II.

CHAPTER IV
RESULTS FOR HOLE-INITIATED AVALANCHE CURRENTS
Table 4-1 lists the experimental data for an RCA diode. The
photocurrent Id generally rises monotonically with the bias voltage
V^. Figure 4-1 shows the measured data of Id/Id V^ = 30 V versus V^
They are in accord with the manufacturer's data which indicates that
no electron-initiated avalanche occurred in these experiments. From
Chapter II we have
ST = S, = 2ql
Id laser ^ eq
(4-1)
Now, let SId = 2 qleqd, Slaser = 2eqleq£, then
!eq = Ieqd + eq*
(4-2)
If the multiplication factor M equals unity, and the primary photocurrent
shows full shot noise, then from Equations (2-10) and (2-12) the follow
ing result is obtained:
ST = ST = 2ql (for M = 1) (4-3)
id V pr
In this case, Ieqd = Ipr. The photocurrent I at the bias voltage of V^ = 30 V; therefore, we can assume Ipr =
I .. with little error. The constant Ie can now be decided by
a vb=30V
29

30
Table 4-1. Experimental data for I = 412 nA
V
Id
!eq
I-eq(-j(I-eq 3,733)
'd/412
W412
30
412
4,145
412
1.000
1.000
100
417
4,159
426
1.012
1.030
150
422
4,180
447
1.024
1.085
170
432
4,268
535
1.049
1.299
185
442
4,480
747
1.073
1.810
200
449
4,637
904
1.090
2.190
240
458
4,722
989
1.112
2.400
260
469
4,792
1,059
1.138
2.570
275
480
5,058
1,325
1.165
3.220
300
492
5,352
1,619
1.194
3.930
320
512
5,887
2,154
1.243
5.230
340
532
6,855
3,122
1.291
7.580

31
Vb=30V
Figure 4-1. Id/Id
vs. Vb for Ipr 412 nA

(4-4)
For the data presented in Table 4-!, IeqA 3 ,733 nA; Igq^ can then be
found by using Equation (4-2).
After obtaining Ieqd> we plot log (Ieqd) versus log (Id). As
discussed in Chapter II, the primary current can then be decided by
comparing this plot with theoretical curves. The Ipr so determined in
Figure 4-2 is exactly equal to what has been assumed, namely Ipr =
The results of Ieq W^pr5 and leq^pr are a^so 9iven
in Table 4-1.
In the data given here, we show the following types of measure
ments: (1) the avalanche photocurrent Id as a function of bias voltage;
(2) the noise, as expressed in the equivalent saturated diode current
Ieqd, as a function of bias voltage; (3) Ieqd as a function of diode
current Id. The measurements are typical of many others. We determined
the diode current Id and the noise Ieqd for different light intensities.
The data for different light intensities are quite similar and generally
consistent with each other as predicted by the theory.
Figures 4-3 to 4-5 show the experimental data obtained for a
primary current Ipr of 412 nA. The Id vs. Vd graph shows a slight bend
which marks the reach-through voltage. The noise is plotted in Figures
4-4 and 4-5 as a function of and Id, respectively.
Table 4-2 lists the data obtained for a different light inten
sity. The Ieq^ for these measurements is obtained by Equation (4-4).
I = 3,962 nA 180 nA = 3,782 nA
eq£
(4-5)

1000
*D
O
V
100
/
/
/
/
/
/
/
/
/
/
/
/
/
101/ 45*
J L
I I 1
-L..-L 1111
10
100
1000
I j ( n A)
Figure 4-2. Log(Ieq^) vs. log(Id) for Ipr
412 nA

(n A)
Figure 4-
550
500
450
400
Reach Through Point
1
100
200
I vs. V, for I = 412 nA
a b pr
GO
L
300
j
vb( v )

(nA)
35
-o
O-
4000
3000
2000
1000
0
vb(v)
0 100 200 300 400
ec>d
Figure 4-4. I
vs. for I = 412 nA

¡qd (n A )
3500
3000
2500
2000
1500
1000
500
200 1
400
450
500
j
Ij (nA)
pr
412 nA
Figure 4-5. 1^ vs. Id for I

Table 4-2. Experimental data for I = 180 nA
V
^eqj^eq 3,782)
Id/180
*eq(j/T80
30
180
3,962
180
1.000
1.00
100
181
4,039
257
1.006
1.43
135
185
4,050
268
1.028
1.49
150
190
4,104
322
1.056
1.79
175
195
4,172
390
1.083
2.17
200
199
4,234
452
1.106
2.51
240
204
4,309
527
1.133
2.93
260
208
4,407
625
1.156
3.47
280
214
4,506
724
1.189
4.02
300
220
4,608
826
1.222
4.59
315
226
4,793
1,011
1.256
5.62
330
231
4,967
1,185
1.283
6.58
340
237
5,217
1,435
1.317
7.97
350
244
5,526
1,744
1.356
9.69

38
which is very close to what we found for previous measurements,
3,733 nA, at a different light intensity. This is in accord with our
assumptions that S, is a constant background noise and independent
of SId.
By comparing the curve of log (Ieq ) versus log (1^) with the
theoretical plots, we note that the curves coincide if the origins
are displaced under 45 by an amount log Ipr, with Ipr = 180 nA, see
Figure 4-6. Figures 4-7 to 4-9 show the results of I. versus V^,
Ieqd versus V^, and Ieqd versus Ij.
Dividing now both axes of Figure 4-5 by the primary current,
we can make a plot of the experimental values
(4-6)
versus
This result is given in Figure 4-10.
The breakpoint structure is not well pronounced, which is due
to the fact that the current gain, or multiplication factor M, for hole-
initiated avalanche current is too small. Therefore, large-voltage
increments are necessary in order to observe current increase while
increasing the bias voltage, and the breakpoint structure could be
easily glossed over. However, we compare the results of Figure 4-10
2
with the theoretical curves of M + var X versus M. The best fit, with
respect to magnitude, is obtained for k = 0.014. The experimental
data plus the theoretical curve is given in Figure 4-11. Comparison of
these experimental results with the asymptotic theory (McIntyre's)
would result in a k value of 0.03, which is too large to believe for a
modern device.

(n A)
39
10
100
1000
Id (n A)
Figure 4-6.
Log(1eqd) vs- 19(Id) for TPr
180 nA

Figure 4-7. 1^ vs. for I
180 nA

pba
41
Figure 4-8. I vs. V, for I = 180 nA
eqd b pr

42
'd(nA)
ec^d
Figure 4-9. I
vs. 1^ for I
180 nA

HO
1 1-1 1.2 1.3
M
Figure 4-10.
Var X + M vs. M for Ipr = 412 nA

X + M
44
Figure 4-11.
Var X + M2
vs. M with theoretical
curve for I = 412 nA
pr

45
2
The experimental M + var X versus M curve obtained from lower
light intensity measurements is presented in Figure 4-12. Comparison
with the theoretical curve for k = 0.014 is made in Figure 4-13. This
diode shows consistent noise performance while operated at different
light intensities.
From the data presented above for low M values and various
light intensities, we conclude that Van Vliet and Rucker's theory is
applicable for hole-initiated avalanche currents.

46
1 1.1 1.2 1.3
Figure 4-12. Var X + vs. M for I
180 nA

47
Figure 4-13.
Var X + M
2
vs. M with theoretical curve for I
pr
180 nA

PART B
EMISSION STATISTICS OF a-PARTICLES

CHAPTER V
THEORETICAL PERSPECTIVES
5.1 Handel1s Theory
It is known that upon scattering a beam of electrons will
emit bremsstrahlung. The power spectrum W(f) of the emitted radiation
is independent of frequency (W = constant) at low frequencies and
decreases to zero at an upper frequency limit f which is approximated
by E/h, where E is the kinetic energy of the electrons, h the Planck
constant. Consequently, the rate of photon emission per unit frequency
interval is N(f) = W/H^, i.e., proportional to 1/f (see Figure 5-1).
Therefore, we conclude that the fraction of electrons scattered with
energy loss e is proportional to 1/e: i.e., the relative squared matrix
element for scattering with energy loss e is [by(e)[ ^ 1/e.
If the incoming beam of electrons is described by a wave func
tion exp[(i/TT) (F. r Et)], the scattered beam will contain a large
nonbremsstrahlung part of amplitude a, and an incoherent mixture of
waves of amplitude aby(e) with bremsstrahlung energy loss e ranging
from some resolution threshold eQ to an upper limit A < E, of the order
of the kinetic energy E of the electrons,
^T
*
d
exp
.TTj
(P r Et)
a
1 +
4
byieje"* £^'/^'de
(5-1)
I I
where by(e) = |by(e)|e has a random phase Y£ which implies incoherence of
2
all bremsstrahlung parts, and |b-j-(e) | is proportional to 1/e.
49

50
w(
Figure 5-
Spectral density of bremsstrahlung in power W and
photon emission rate N

In Equation (5-1) the frequency-shifted components present in
the integral interfere with the elastic term, yielding beats of fre
quency e/tr. The particle density given by Equation (5-1) is
kT|2 = |a2l
1 + 2
j
rA
Iby(e)
cos
f '
+ y
tr e
£0
J
A
b*(e)b-j.(c' Je1' e^^dede'
'0 &0
(5-2)
the second term in large parentheses describes the particle density
beats.
If the particle concentration fluctuation is defined by
2 2 2
6 |xp| = |^| <|ijj| >, its autocorrelation function will be
<<5Ht5Mt+x> = 2
2 a
'A
|b(e)( cos
ex
de
(5-3)
2
which is proportional to |b(e)| and hence proportional to 1/f. There
fore, the spectral density of the particle concentration fluctuation
[the Fourier transform of Equation (5-3)] is proportional to 1/f.
The relative bremsstrahlung rate |b(e)| can be derived as
follows. The constant spectral energy density can be written as
22 3
oo(f) = 4e (AV) /3c k, where e is electronic charge, c is the velocity of
light, < is the dielectric constant of the medium, and AV is the

velocity change in the scattering process. The relative scattering rate
2
density with energy loss e, |b(e)| is obtained by dividing w(f) by the
energy of a photon e = hf:
|b(f)|2 = 4efiAVli = ml-fL (5-4)
3c32,rtrf,c Tc 37rfK fK
|b(S)l2 = 4e£(AV)i = CCA (5-5)
3c3he< eK
where
A = 2(AB)2 .w = AV
A 3tt l c
e2
and a = is the fine structure constant. In the MKS system, a =
Tvc
where Kq is the dielectric constant of vacuum. The spectral den
sity of the relative fluctuations is
(5-6)
TTckq
S ^ = 2[1 + aA An (A/e)] ^aA/fic
kl
2aA/fx
(5-7)
5.2 The Allan Variance Theorem
The main link between counting statistics and particle current
noise is provided by MacDonald's theorem,
d xahA 1
7T
r
S (co)oj ^sin uTdc
, m
(5-8)

53
with inversion
0
(5-9)
where is the variance of the total number of particles detected in
a time interval (t, t + T) and Sm(w) the noise spectral density of the
flux fluctuations Am(t). This theorem is useful for Poissonian statis
tics. Unfortunately, for 1/f noise, Equation (5-8) is not applicable,
since the integral diverges. However, a useful concept in this case is
the "two sample variance" or "Allan variance." Let mj^ be the average
(2)
counting rate in (t, t + T) and the counting rate in (t + T, t + 2T)
Then the Allan variance is defined by
(5-10)
and
(5-11)
A? A 2
The variance a [which means (o ) ] turns out to be finite for
1/f noise.
The theorem reads [15]
(5-12)
with inversion

54
S
m
2ni
i 00+3
i oo+3
C
dP
eos 2 Ptt
1 2
P-3
r(P)
dT A2m
P aM
OT "t
(5-13)
For Poissonian shot noise S (w) = 2mo, where mQ is the average counting
a?
rate. Substituting Sm(o)) into Equation (5-12), one has 0^(1) = mQT.
2ttC
For 1/f noise, with a spectrum of Sm(w) = where C is a constant,
A2 2
the Allan variance cty| = 2CT £n 2. The various results are summarized
in Table 5-1.
5.3 Application of the Allan Variance Theorem
to Counting Statistics
The presence of 1/f noise in counting statistics can now be
determined from a measurement of the Allan variance as a function of T.
For suppose that the noise is composed of shot noise and 1/f noise, i.e.,
aJ2(T) = mQT + 2CT2 £n 2 (5-14)
recall that = m T, then a measurement of the relative Allan variance
T o
R(T) = a^2(T)/2 yields
R(T) = + 2C £n 2 (5-15)
o
2
where C = C/m is a constant. For short-time intervals the term
o
1/m T is dominant, hence R(T) is proportional to 1/T. When T is long
enough, 2C £n 2 becomes dominant; R(T) is, therefore, a constant,
called the "flicker floor."

55
Table 5-1. Various
results of Allan
vari anee
theorem
Sm(>
Mt(T>
Poissonian
2mo
= mQT
shot noise
General
shot noise
2 = Km T
T o
1/f noise
2ttC / | co |
2CT2 In 2
Lorentzian
flicker noise
4B a
4b 2 2
B
2
[4e~aT e'2aT + 2aT 3]
a + a)
a
Pathological
noise
L/M^
0 LTX(1 2X"2)
sin (iTA/2)r(A + 1)

bb
According to Handel's theory, the constant C in the 1/f noise
term in the relative Allan variance is
C = 2oA£/k (5-16)
where 5 is a coherence factor; for a-particles it is expected to be close
2 2
to one; a is the fine structure constant 1/137, and A = 2(Av) /3ttc
where Av is the velocity change of the particles in the emission process,
c is the velocity of light. For a-particles one finds
= 8.32 x 10"7 x -L (5-17)
< K
where E is energy in MeV and k is the dielectric constant of the radio
active material.
In these experiments E = 5.48 MeV, so that the value of 2C £n 2
(the value of the flicker floor, F) is
F = 4? 8~32 x 7 x 5-4-8- x jn 2 126.4 x 107 x -£-
K K
K
(5-18)

CHAPTER VI
EXPERIMENTAL SET-UP FOR LONG-TIME COUNTING
The block diagram of the counting system being used to investi
gate 1/f fluctuations in the a-particle emission rate is shown in
241
Figure 6-1. The source is g^Am which decays with a half-life time
of T-j/2 = 458 years with the emission of 5.48 MeV a-particles into
237
g2Np The detector, a silicon surface-barrier detector, is reverse
biased at 80 volts, and the dead times of the ND575 Analog-to-Digital
Converter (ADC) and ND66 Multi-Channel Analyzer are 60 n-seconds and
6 y-seconds, respectively. Therefore, no dead-time correction is neces
sary, as long as the counting rate is kept lower than 1,000 counts per
second [16] (or the averaged time elapse between two counts is higher
than 1,000 y-seconds).
A typical full-energy spectrum measured in these experiments
is shown in Figure 6-2 in semi-logarithmic scale. The spectrum is
shown on a display screen while accumulating counts and the final
result, after a chosen time T, is stored in the memory units of the ND66.
The Full Width Half Maximum (FWHF) of the spectrum can be found by
moving the cursor on the display, which indicates the number of counts
under each energy channel. The systematic range used in these experi
ments was from peak channel -FWHM x 6 to peak channel +FWHM x 2. There
fore, the total number of counts M-j., which will be analyzed later, is
always under a fixed portion of the full spectrum.
57

Figure 6-1. Block diagram counting system

NUMBER OF COUNTS
Figure 6-2. Typical full-energy spectrum

60
The M-p's of adjacent time intervals can be read directly from
the memory units of the ND66 Multi-Channel Analyzer; thus the Allan
variance can be calculated by
1
2
1
N 1
N-l
l
i = 1
- M
Ci+D]
T
(6-1)
since
1 N m
the relative Allan variance, R(T), defined by Equation (5-15), can be
found by using Equations (6-1) and (6-2).
Part of the data, mostly the total number of counts for T longer
than 1,000 minutes, was not read directly from the ND66 Multi-Channel
Analyzer. An "add-up" method was adopted; namely, m|qqq was obtained by
(1) (2) (2) (3) (4)
adding up M£qq and M£qq, and M^qqq equals the sum of M£qq and M£qq, etc.
Physically, since M^qS were measured in adjacent time intervals, of
course the first two can be added up as the total number of counts for
the first 1,000-minute interval, and the third and the fourth can be
(21
summed up as M]qqq- However, in order to check the validity of this
method, the following experiment has been done.
By making use of a T-connector, the output signal of the Ortec
410 amplifier was fed simultaneously into two ND575 ADCs (see Figure
6-1). The first one (ADC #1) counted 100-minute measuremenst for 310
times, and the second one (ADC #2) accumulated 500-minute counts for 62

times; hence, both ADCs covered exactly the same time span. The "add-
up" method was applied to M-jqqS, obtained from ADC #1, to find out the
calculated M^qS.
Table 6-1 lists the results calculated from both ADCs for
500-minute measurements. The difference between them, in each category,
is <1%. This shows the validity of the "add-up" method.
Table 6-1. Comparison of the results obtained
from the "add-up" method (ADC #1)
and the real-time measurements
(ADC #2)
ADC #1
ADC #2
A
o
o
LO
9,062,464.36
9,067,009.84
A2
0^5OO
7,862,171.3
7,943,285.5
R(500)
9.573 x 10-8
9.662 x 10'8

CHAPTER VII
COUNTING RESULTS
7.1 Convergence of the Allan Variance for N ->
Because the existence of a "variance of the variance" (or, a
noise of the noise), the relative Allan variance itself is a fluctuat
ing parameter. In order to obtain an accurate value of R(T), a suf
ficient number of measurements must be made especially when T is short.
To determine the minimum number of measurements needed, we plot R(T)
versus N for different T's. These figures show that when N is small,
R(T) is spread over a wide range. When T is increased the spread in
R(T) diminishes and finally R(T) converts to a stable value. For
example, Figure 7-1 shows that for T = 100 minutes, 21 measurements of
M(100) yield an R(100) = (7.09 3.94) x 107; for N = 24, R(T) =
(7.23 2.14) x 107; for N = 27, R(T) = (7.13 1.85) x 10-7. Here
R(T) is given in the form of (mean value standard deviation). From
Figure 7-2 we know that for T = 1 minute at least 70 measurements are
necessary for a reliable value of R(T). Figure 7-3 shows that for
T = 3 minutes we need N ^ 50.
7.2 Results of the Relative Allan Variance versus 1/T
From Equation (5-15) we have
R(T) = + 2C' £n 2 (7-1)
62

63
40
30
rs.
20
10
o i 1 i L
10 20 30
Figure 7-1. R(T) vs. N for T = 100 min.

N
Figure 7-2 R(T) vs. N vor T = 1 min

OIxcoa
Figure 7-3. R(T) vs. N for T = 3 min.

where C is a constant, mQ is the averaged counting rate, and T is
the counting time.
Figures 7-4 to 7-10 show seven series of experimental results
of R(T) versus 1/T. Figures 7-4 to 7-6 show clearly that for T small
(1/T large), R(T) is equal to the value of Poissonian shot noise, the
first term of Equation 7-1. But for T large (1/T small), the values
of R(T) is larger than the value for shot noise. Figures 7-7 to 7-10
show the results of some experiments which were concentrated on long
time measurements. They all show that for T long enough the experimental
values of R(T) are higher than l/mQT and the curves start to level off,
which implies the existence of 1/f noise. The second, constant term in
Equation (7-1) becomes dominant.
7.3 Results of the Allan Variance versus T
A 2
Figures 7-11 and 7-12 give the plot of the Allan variance, aM ,
versus T corresponding to the measurements of Figures 7-4 and 7-5,
A2
respectively. Recall that for shot noise = mQT, which is propor-
A2 2
tional to T, but for 1/f noise qv|y = 2CT £n 2, which is proportional
2
to r.
Since these figures (Figures 7-11 and 7-12) are plotted on
log-log scale, the slope of log (o$y) versus log(T) is really the
A2
exponent of T in the expression of a^y, see Equation (5-14). Figures
A2 2
7-11 and 7-12 show clearly the dependence of c^y on T and T while
different noise terms dominate the noise spectrum.

Rexp(T)
67
Figure 7-4. Rexp(T) vs. 1/T for m0 = 27,700/min.

68
Figure 7-5.
R (T) vs. 1/T for m = 17,980/min.
6Xp O

exp
Poisson Noise
-4
10
O Exp. Result
-5
10

a
q/3
7

ac
/
-6
10
: /
-7
10
7
0 /
r
1 1 1 L-
-3 -2 -1
10 10 10 i
(min.-1 )
Figure 7-6.
R (T) vs. 1/T for m = 18,160/min.
c A [J U

exp
70
Figure
7-7. R (T) vs. 1/T for m = 17,180/min.
exp o

71
Figure 7-8. Rexp(T) vs. 1/T for mQ = 18,356/min.

dxe
Poisson Noise
0 Exp. Result
Figure 7-9. R (T) vs. 1/T for m 17,807/min.
GXp U

73
Figure 7-10. RgXp(T) vs. 1/T for mo 18,180/min.

74
A2
Figure 7-11. aM (T) vs. T for rn = 27,700/min.
' 0

75
T (m in.)
(T)
vs.
T for m =
o
Figure 7-12.
17,960/min.

CHAPTER VIII
DISCUSSION OF RESULTS
The curves shown in Chapter VII are not smooth, which is due
to the nature of the noise fluctuations. In order to obtain a univer
sal curve, the following procedure has been carried out.
Assume the experimental results of R(T) contain a fluctuation
term AR, due to the presence of "noise of noise," i.e.,
Rexp(T) = iTT+ 2C £n 2 + AR (8-1)
^ o
where the average value of AR, , should be zero. The experiment was
then repeated for several times, and the average of ReXp(T),
= HTT + 2C' £n 2 + (8-2)
is obtained. Although the value of in Equation (8-2) can hardly be
exactly zero for a finite number of measurements, it should be reduced
by a great deal compared with the value of AR for a single measurement.
Now, the value of gives the best estimation to the true value
of R(T).
The distance between the radioactive source and the detector was
adjusted such that very close counting rates were obtained while
repeating the measurements. However, it is difficult to obtain identical
76

counting rates. Therefore, the shot noise term in the relative Allan
variance, R(T), is slightly different for each series of measure
ments .
Before the average of Rexp(T) is taken, the shot noise term,
¡yy should be normalized to the same counting rate. Here a rate of
o
18,000 counts per minute was chosen. The normalized relative Allan
variance R (T) is then
Rn^ = Rexp^ mV + 18,000 x T = 18,000 x T + 2C' 2
(8-3)
Tables 8-1 and 8-2 show the values of ReXp(T) and Rn(T). Instead
of <'ReXp(i)> is now used to estimate the true value of R(T).
The value of is calculated as follows:
ZR .(T) x DF
= ni 1
ZDF.
(8-4)
f"h
where R .(T) is the value of R (T) obtained from the i series and DF.
m n i
stands for the degrees of freedom of that particular value, which equals
the number of measurements minus one.
Figure 8-1 shows the comparison of and (1/18,000 x T)
+ 1 x 10-^ versus 1/T, which suggests that the value of 2C £n 2 is
about 1 x 10"^.
Figure 8-2 shows the comparison of the average value of the
A2
normalized Allan variance, , which is obtained by

Table 8-1.
The values
being the i
of Rexp(T) anc* Rn(T) fr T =
number of measurements
1-50, all values x 10 the
numbers in
parentheses
Time (min.)
m
0
1
2
5
10
20
50
27,700/min.
Rexp
Rn
348.7
544.1(96)
205.3
303.0(48)
84.27
123.30(19)
39.07
58.58(38)
11.91
21.66(18)
4.957
8.844(7)
17,980/min.
WT>
Rn
572.9
572.5(177)
232.1
231.9(88)
149.6
149.6(93)
41.16
42.78(83)
22.68
23,49(41)
11.65
11.970(16)
18,160/min.
Rexp
Rn
504.8
509.8(248)
263.1
265.6(124)
71.85
72.85(49)
66,03
66.55(88)
34.11
34.37(44)
11.350
11.450(149)
17,807/min.
Rexp
Rn'T>
12.150
12.030(56)

Table 8-2. The values of ReXn(T) ancl Rn(T) for T =
ses being the number of measurements
Time (min.)
m
0
100
200
Rpvn(T)
4.126
3.281
27,770/min.
exp
R(T)
6.075(18)
4.255(9)
Rovn(T)
6.359
4.093
17,980/min.
exp
R (T)
nv
6.348(72)
4.087(36)
R n(T)
7.398
3.505
18,160/min.
exp
R(T)
7.437(310)
3.524(155)
R.yn(T)
7.192
8.390
17,180/min.
exp
Rn
6.927(26)
8.257(13)
Rpvn(T)
6.003
1.970
18,356/min.
exp
Rn 6.111(30)
2.024(15)
100-3,000, all values x 10-^, the numbers in parenthe-
500
1,000
2,000
6.105
6.494(3)
3.404
6.8300
3.401(14)
6.8290(7)
0.9662
0.8133
1.3830
0.9742(62)
0.8175(31)
1.3850(15)
8.9990
8.9460(5)
2.3430
4.4280
2.3640(6)
4.4390(3)
3,000
3.360
2.3610(10)

Table 8-2Continued
Time (min.)
m
0
100
200
R n(T)
8.147
4.263
17,807/min.
exp
Rn
8.087(28)
4.233(14)
R nlD
3.808
18,180/min.
exp
Rn
3.836(40)
500
1,000
2,000
5,000
2.3630
2.3510(5)
1.3200
0.8042
0.6299
0.5865
1.3310(30)
0.8099(15)
0.6327(7)
0.5884(5)

Page
Missing
or
Unavailable

NORMALIZED ALLAN VARIANCE
A?
Figure 8-2. vs. T for mQ = 18,000/min.

83
= x ^8,000 x T)2 (8-5)
Very good agreement is obtained as shown in this figure.
The value of and 1/18,000 x T (shot noise level) are
listed in Table 8-3. It shows clearly that at this counting rate,
18,000 counts per minute, for T longer than 100 minutes, 1/f noise
becomes noticeable. For T longer than 1,000 minutes, 1/f noise totally
dominates the noise spectrum.
Besides the fluctuation of scattering cross section described
in Handel's theory, two other possibilities may contribute to the
fluctuation in the total number of counts. First, the pulse height,
which is produced when an a-particle is absorbed by the detector, is a
strong function of the bias voltage applied to the detector [17]. If
the bias voltage was unstable, then a-particles with the same energy
would have been registered in different energy channels. Fortunately,
the detector control unit used in these experiments, Ortec 210, has a
very good stability: bias voltage variation with line voltage is
<0.005% for 105-125 V AC input, and the stability is 0.01% [18]. Secondly,
since a fixed portion of the full spectrum is counted, whether particles
with energies around boundaries fall in or out of boundaries contributes
fluctuations to the total number of counts.
Therefore, a wider energy range should have less fluctuations
in the total number of counts in a certain time period. This is indeed
the case; see Table 8-4. However, for T = 100 minutes, the difference
between R(T)'s for wider range and narrower range is negligible. For

Table 8-3.
The values
of (T)> and
1/18,000 xT,
all values
x 10"7
Time (min.)
1
2
5
10
20
50
537.4
261.0
123.3
55.66
27.86
11.56
1
18,000 x T
555.6
277.8
111.1
55.56
27.78
11.11
Time (min.)
100
200
500
1 ,000
2,000
3,000
7.156
3.825
1.799
1.6480
1.1590
1.8160
1
18,000 x T
5.556
2.778
1.111
0.5556
0.2778
0.1852

Table 8-4. Comparison of experimental results for narrower energy
range (peak channelFWHM x 6 to peak channel + FWHM x 2)
and wider energy range (peak channelFWHM x 12 to peak
channel + FWHM x 6)
Time (min.)
Channels 127-147a
Channels 112-1573
o
o
V
1,812,492.94
1,828,630.89
100
A2
M100
2,430,286.7
2,472,416.52
R(100)
7.398 x 107
7.394 x 10'7
A
o
o
LD
5
9,067,009.8
9,147,489.02
500
A2
G|V|500
7,943,285.5
7,762,255.74
R(500)
9.662 x 10"8
9.277 x 10"8
A
o
o
oo
5
54,398,712.1
54,881,014.1
3,000
A2
aM
"3000
698,293,362
634,854,514
R(3000)
2.360 x IQ"7
2.108 x 10-7
aPeak channel: 142; FWHM: 2.5 channels

86
T = 50 minutes, the difference is 4%, but the value of is
n
62% higher than the shot noise level (see Table 8-3). For T = 3,000
minutes, the difference is 11%, and the value of is ten
times the value for the shot noise level. Hence the energy range
chosen in these experiments affected little the final results.
In certain R(T) versus 1/T figures in Chapter VII and in
Figure 8-1, when T becomes large R(T) shows the tendency to be propor
tional to T. This phenomenon is not exclusive, since according to the
Allan variance theorem R(T) is indeed proportional to T for very slow
Lorentzian flicker noise [15].

CHAPTER IX
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
From the experimental data presented in the Appendix, we see
that the ratio for the averaged value of the total number of counts,
, to the variance of M is very close to one when T is less than
100 minutes, which is an indication for Poissonian statistics. However,
when T is greater than 100 minutes this ratio becomes greater than
unity and generally increases monotonically with T, which indicates
super-Poisonian statistics. With the help of the newly devised Allan
variance theorem, we found that this excess noise in the radioactive
decay rate has a 1/f spectrum.
Quantitatively, from Equation (5-18), we know the flicker floor
F has a value of
F = 126.4 x 10'7 x (9-1)
K
To the author's knowledge, nobody has ever measured the dielectric
constant of Am02, the a-particle source used in these experiments; thus
the value of k in Equation (9-1) is unknown. However, since the atomic
structure of Am02 is similar to that for U02, the dielectric constant
of U02 (20.4 1.5 [19], 21.7 0.5 [20], and 21.0 1 [21]) can then
be used as a reference.
If one considers the following factors that (1) the coherence
factor, C, is always less than unity; (2) the dielectric constant of
87

88
Am02 may be larger than 21, then the measured value of F = 1 x 10
is in the right ballpark to verify Handel's theory. In particular,
for c = 0.17, k = 21, one obtains F = 1.02 x 10'., in accord with the
observed value of Figures 8-1 and 8-2. We believe, therefore, that
these experiments constitute the first experimental verification of
Handel's quantum 1/f noise theory.
241
The data presented here all referred to g^Am If an a-source
with different energy peaks is used, the counting could then be done
in each of these energy peaks. This should result in an R(T) versus 1/T
as given in Figure 9-1.
If Handel's theory is correct, the flicker floor should be
2 2
proportional to v /c i.e., to the energy E. Thus, the flicker floors
shown in Figure 9-1 should have the ratios of these energy peaks. This
experiment will therefore be a very strong test to verify the appli
cability and correctness of Handel's theory.

89
T
Figure 9-1. R(T) vs. 1/T for different energy peaks

APPENDIX
EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS

Table A-l. Experimental data for mQ 27,700
T
N

var.
var

A2
aMT
WT)
1
96
27,761
29,759
1.07
26,874
3.487 x 10'5
2
48
55,525
64,254
1.16
63,300
2.053 x 10"5
5
19
138,797
174,979
1.26
162,334
8.427 x 10"6
10
38
277,369
271,146
0.98
300,625
3.907 x 106
20
18
554,689
407,757
0.74
366,675
1.191 x 106
50
7
1,386,606
1,556,691
0.83
950,741
4.945 x 10"7
100
18
2,772,026
13,322,534
4.81
3,170,123
4.126 x 10"7
200
9
5,544,053
49,414,157
8.91
10,084,511
3.281 x 10-7
500
3
13,855,521
202,833,876
14.64
117,194,752
6.105 x 10~7

Table A-2. Experimental data for mQ 17,980
T
N

var.
var.

A2
aMj
R exp
1
177
17,987
17,155
0.95
18,534
5.729 x 10'5
2
88
35,970
29,055
0.81
30,030
2.321 x 10"5
5
93
89,984
108,003
1.20
121,148
1.496 x 10"5
10
83
185,379
154,777
0.84
141,464
4.116 x 106
20
41
370,752
324,936
0.88
311,757
2.268 x 10'6
50
16
926,836
868,989
0.94
1,000,977
1.165 x 10"6
100
72
1 ,796,155
15,133,266
8.43
2,051,505
6.359 x 107
200
36
3,592,309
55,659,746
15.49
5,282,260
4.093 x 10-7
500
14
8,980,092
324,379,609
36.12
27,448,241
3.404 x 10'7
1,000
7
17,960,183
1,311,457,070
73.02
220,310,187
6.830 x 107

Table A-3. Experimental data for mQ 18,160
T
N

var
ya.r.

A2
aMj
R (T)
exp
1
248
18,160
16,829
0.93
16,647
5.048 x 10-5
2
124
36,321
34,706
0.96
34,714
2.631 x 10'5
5
49
90,795
64,814
0.71
59,228
7.185 x 10"6
10
88
181,684
214,732
1.18
217,969
6.603 x 10'6
20
44
363,367
436,054
1.20
450,422
3.411 x 10-6
50
149
908,647
1 ,274,359
1.40
936,880
1.135 x 106
100
310
1,812,493
9,116,358
5.03
2,430,287
7.398 x 107
200
155
3,625,004
31 ,272,439
8.63
4,606,285
3.505 x 10"7
500
62
9,067,010
168,503,103
16.58
7,943,286
9.662 x 10'8
1,000
31
18,134,020
668,501 ,571
36.86
26,745,242
8.133 x 10-8
2,000
15
36,265,809
2,626,291,950
72.42
181,946,661
1.383 x 10-7
3,000
10
54,398,714
5,991 ,195,130
110.14
698,293,362
2.360 x IQ*7

Table A-4. Experimental data for mQ 17,180
T
N

var
var

A2
aMj
R (T)
exp
100
26
1,717,908
10,533,131
6.13
2,122,404
7.192 x 107
200
13
3,435,815
38,770,910
11.28
9,904,640
8.390 x 10'7
-7
500
5
8,589,200
186,528,449
21.72
66,392,064
8.999 x 10
UD
4^>

Table A-5. Experimental data for mQ 18,356
T
N

var
var

A2
aMT
R (T)
exp
100
30
1,835,679
3,795,975
2.07
2,022,965
6.003 x 10"7
200
15
3,671,362
10,830,882
2.95
2,654,939
1.970 x 10'7
500
6
9,178,393
64,397,080
7.02
19,739,443
2.343 x 10-7
1,000
3
18,356,785
295,307,233
16.09
149,225,472
4.428 x 10"7
cn

Table A-6. Experimental data for mQ 17,807
T
N

var
var

A2
aMT
WT>
50
56
890,370
1,298,893
1.46
963,275
1.123 x 10"6
100
14
1,780,739
3,446,788
1.94
2,583,343
5.616 x 10"7
200
14
3,561,478
8,551 ,758
2.40
5,407,350
2.808 x 10'7
500
5
8,902,655
25,397,533
2.85
18,731,008
1.123 x 10'7

Table A-7.
Experimental
data for m
0
18,180
T
N

var.
var

0%
Rexp
200
40
3,636,682
5,835,231
1.61
5,036,452
3.808 x 10'7
500
30
9,093,104
18,423,928
2.03
10,914,004
1.320 x 10"7
1,000
15
18,186,209
60,223,202
3.31
26,598,254
8.042 x 10"8
2,000
7
36,369,629
92,776,209
2.55
83,318,101
6.299 x 108
3,000
5
54,558,626
338,426,463
6.20
174,587,904
5.865 x 10"8

REFERENCES
[1] A. S. Tager, "Current fluctuations in a semiconductor (dielec
tric) under the conditions of impact ionization and avalanche
breakdown," Sov. Phys.Solid State, Vol. 6, pp. 1919-1925, 1965.
[2] R. J. McIntyre, "Multiplication noise in uniform avalance diodes,"
IEEE Trans. Electron Devices, Vol. ED-13, pp. 164-168, 1966.
[3] R. J. McIntyre, "The distribution of gains in uniformly multi
plying avalanche photodiodes: Theory," IEEE Trans. Electron
Devices, Vol. ED-19, pp. 167-190, 1971.
[4] S. D. Personick, "New results on avalanche multiplication statis
tics with applications to optical detection," Bell Syst. Tech. J.,
Vol. 50, pp. 167-190, 1971.
[5] S. D. Personick, "Statistics of a general class of avalanche
detectors with applications to optical communication," Bell Syst.
Tech. J., Vol. 50, pp. 3075-3094, 1971.
[6] G. E. Stilman and C. M. Wolfe, "Avalanche photodiodes," in
Semiconductors and Semimetals, Vol. 12. R. K. Willardson and
A. C. Baer, Eds. New York: Academic Press, 1979, Ch. 5.
[7] K. M. van Vliet and L. M. Rucker, "Theory of carrier multiplication
and noise in avalanche devicesPart I: One-carrier processes,"
IEEE Trans. Electron Devices, Vol. ED-26, pp. 746-751, 1979.
[8] K. M. van Vliet and L. M. Rucker, "Theory of carrier multiplication
and noise in avalanche devicesPart II: Two-carrier processes,"
IEEE Trans. Electron Devices, Vol. ED-26, pp. 752-764, 1979.
[9] J. Gong, K. M. Van Vliet, A. D. Sutherland, and E. R. Chenette,
"Noise measurements on photo avalanche diodes," Phys. Stat.
Sol. (a) 63, pp. 445-460, 1981.
[10] P. H. Handel, "Quantum approach to 1/f noise," Physical Review
(A), Vol. 22, No. 2, pp. 745-757, 1980.
[11] K. L. Ngai, "A unified theory of 1/f noise and dielectric
response in condensed matter." Proceedings of second inter
national symposium on 1/f noise, pp. 445-476, 1980.
98.

99
[12] S. Machlup and T. Hoshiko, "Scale invariance implies 1/f spec
trum," Proceedings of second international symposium on 1/f
noise, pp. 556-558, 1980.
[13] C. M. Van Vliet, A. van der Ziel, and P. H. Handel, "Super-
statistical emission noise," Physica 108A, pp. 511-526, 1981.
[14] W. Lukaszek, A. van der Ziel, and E. R. Chenette, "Investigation
of the transition from tunneling to impact ionization multipli
cation in silicon p-n junctions," Solid-State Electron., Vol. 19,
pp. 57-71, 1976.
[15] C. M. Van Vliet and P. H. Handel, "A new transform theory for
stochastic processes with special application to counting
statistics," Physica 113A, pp. 261-276, 1982.
[16] G. F. Knoll, "Radiation detection and measurement," New York, Wiley
and Sons, p, 95, 1979.
[17] W. J. Price, "Nuclear radiation detection," New York, McGraw-Hill,
p. 273, 1958.
[18] Ortec technical data, 210 detector control, 1971.
[19] A. D. B. Woods, G. Dolling, and R. A. Cowley, "The crystal
dynamics of uranium dioxide," Inelastic scattering Neutrons,
Proc. Symp., 4th, Bombay, pp. 373-378, 1964.
[20] D. J. Huntley, "The dielectric constant of U02 and its variation
with porosity," Can. J. Phys. 44(11), pp. 2952-2956, 1966.
K. Gesi and J. Tateno, "Dielectric constant of uranium dioxide
at 9.4 GH2," Jap. J. Appl. Phys., 8(11), pp..1358-1359, 1969.
[21]

BIOGRAPHICAL SKETCH
Jeng Gong was born in Taipei, Republic of China, on July 18,
1953. He attended the National Cheng Kung University in Tainan,
Taiwan, and received the Bachelor of Science degree in electrical
engineering in 1975. After entering the Chinese Army from 1975 to 1977,
he worked with Dah-Shen Electronic Company, Taipei, until 1978.
Entering the University of Florida in 1979, he received the
Master of Engineering degree in electrical engineering in 1980. From
1981 to the present, he has pursued his work toward the degree of
Doctor of Philosophy.
In 1978, he married the former Chang-Ling Tseo and is the
father of one son, Dow.
100

I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
/vV l/isu l
Carolyn M. Van Vliet, Chairperson
Professor of Electrical Engineering
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
C/$
A. van der Ziel ^
Graduate Research Professor of
Electrical Engineering
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
A. D, Sutherland
Professor of Electrical Engineering
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
R_ cA^
E. R. Chenette
Professor of Electrical Engineering

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Associate Professor of Nuclear
Engineering Science
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
P. H. Handel
Professor of Physics
University of Missouri
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August 1983
. /$
Dean, College of Engineering
Dean for Graduate Studies and Research



Rexp(T)
67
Figure 7-4. Rexp(T) vs. 1/T for m0 = 27,700/min.


42
'd(nA)
ec^d
Figure 4-9. I
vs. 1^ for I
180 nA


8
The asymptotic limit for N -> is not trivial; in the paper of van
Vliet and Rucker it is shown that (2-7) and (2-8) lead to McIntyre's
results.
For hole initiated avalanche case, replace k by 1/k, X by y,
and y by X. The results are
Mt¡ E fc =
(1
kN(l
k)[k(l
+ u)T
+ ur (k + y)
N+T
(2-9)
var
<(axJ¡)2> =
- DO k)
2k + yk + y
k-u2
k + y
x
" 1
1 + V
M
h 1
juni
- kj|
(2-10)
We now consider Equations (2-9) and (2-10) in more detail. At
the onset of ionization the field is just high enough to sustain one
possible ionization in the avalanche region; thus N = 1. With
increasing field, the value of y, denoted as y(l), increases and so
the gain increases according to Equation (2-9). The length £-j neces
sary to gain the ionizing energy simultaneously decreases, until two
ionizations per carrier transit are possible. The value of y(l),
just prior to this is denoted as uO)max- When two ionizations per
carrier transit are possible, the diode switches over from the regime
N = 1 to the regime N = 2. To realize the gain M of that operating
point, the value of y for the new regime, denoted as y(2), is consider
ably less than the value of y(l)m=v prior to the switch-over, as is
found by inverting (2-9) for y with M fixed, taking N = 1 and N = 2,


60
The M-p's of adjacent time intervals can be read directly from
the memory units of the ND66 Multi-Channel Analyzer; thus the Allan
variance can be calculated by
1
2
1
N 1
N-l
l
i = 1
- M
Ci+D]
T
(6-1)
since
1 N m
the relative Allan variance, R(T), defined by Equation (5-15), can be
found by using Equations (6-1) and (6-2).
Part of the data, mostly the total number of counts for T longer
than 1,000 minutes, was not read directly from the ND66 Multi-Channel
Analyzer. An "add-up" method was adopted; namely, m|qqq was obtained by
(1) (2) (2) (3) (4)
adding up M£qq and M£qq, and M^qqq equals the sum of M£qq and M£qq, etc.
Physically, since M^qS were measured in adjacent time intervals, of
course the first two can be added up as the total number of counts for
the first 1,000-minute interval, and the third and the fourth can be
(21
summed up as M]qqq- However, in order to check the validity of this
method, the following experiment has been done.
By making use of a T-connector, the output signal of the Ortec
410 amplifier was fed simultaneously into two ND575 ADCs (see Figure
6-1). The first one (ADC #1) counted 100-minute measuremenst for 310
times, and the second one (ADC #2) accumulated 500-minute counts for 62


ACKNOWLEDGMENTS
I am deeply indebted to Dr. Carolyn Van Vliet, Dr. Aldert
van der Ziel, and Dr. Peter Handel for their most generous and valuable
guidance and assistance during the preparation of this work. I thank
Dr. Alan Sutherland, Dr. Eugene Chenette, and Dr. Gys Bosman for their
advice and encouragement. Dr. Wiliam Ellis has been most helpful
during many phases of this work and his help, especially in discussions
of radiation detection techniques, is greatly appreciated.
I also wish to express my appreciation to Dr. R. J. McIntyre
and Dr. Dean Schoenfeld for providing RCA diodes and the Argon-ion
Laser, respectively. I am particularly indebted to Mrs. S. L. Wang for
the preparation of the figures presented in this work.
My deepest gratitude goes to my parents, my wife, and my son
who have always had faith, encouragement, and understanding when it
was most needed.
This research was supported by AFOSR contract, #82-0226.
11


CHAPTER II
THEORY OF NOISE IN AVALANCHE DIODES
The first theories for noise in avalanche diodes were given
by Tager (1965) [1] and by McIntyre (1966) [2]. Tager assumed that
the ionization coefficients for ionization by electrons and holes are
equal. Under these conditions the noise spectral density is given by
St = 2ql M3 (2-1)
ld p
where 1^ is the avalanche current, I is the primary photocurrent,
q is the electronic charge, M is the average gain, and where it is
assumed that the incoming particles show full shot noise.
The theory by McIntyre allowed for the possibility that the
ionization coefficients are unequal. Let a(x)dx be the probability
for ionization by an electron in the interval (x,x + dx) and let g(x)
be the ionization probability by a hole. For simplicity, McIntyre
assumed B(x) = ka(x). The noise spectral density is then given by
S
2a I M'
pr
1 (1 k)
M l'
2-t
M
/
(2-2)
for the case that the primary carriers injected into the avalanche
region are electrons; in case these carriers are holes, replace k by
1/k; if both occur, add the two partial results.
5


63
40
30
rs.
20
10
o i 1 i L
10 20 30
Figure 7-1. R(T) vs. N for T = 100 min.


46
1 1.1 1.2 1.3
Figure 4-12. Var X + vs. M for I
180 nA


13
2
computer plots for M + var X versus M are obtained from (2-9) and
(2-10). The results are given in log-log form in Figures 2-2 to 2-4
and on linear paper in Figures 2-5 to 2-7.


53
with inversion
0
(5-9)
where is the variance of the total number of particles detected in
a time interval (t, t + T) and Sm(w) the noise spectral density of the
flux fluctuations Am(t). This theorem is useful for Poissonian statis
tics. Unfortunately, for 1/f noise, Equation (5-8) is not applicable,
since the integral diverges. However, a useful concept in this case is
the "two sample variance" or "Allan variance." Let mj^ be the average
(2)
counting rate in (t, t + T) and the counting rate in (t + T, t + 2T)
Then the Allan variance is defined by
(5-10)
and
(5-11)
A? A 2
The variance a [which means (o ) ] turns out to be finite for
1/f noise.
The theorem reads [15]
(5-12)
with inversion


CHAPTER IV
RESULTS FOR HOLE-INITIATED AVALANCHE CURRENTS
Table 4-1 lists the experimental data for an RCA diode. The
photocurrent Id generally rises monotonically with the bias voltage
V^. Figure 4-1 shows the measured data of Id/Id V^ = 30 V versus V^
They are in accord with the manufacturer's data which indicates that
no electron-initiated avalanche occurred in these experiments. From
Chapter II we have
ST = S, = 2ql
Id laser ^ eq
(4-1)
Now, let SId = 2 qleqd, Slaser = 2eqleq£, then
!eq = Ieqd + eq*
(4-2)
If the multiplication factor M equals unity, and the primary photocurrent
shows full shot noise, then from Equations (2-10) and (2-12) the follow
ing result is obtained:
ST = ST = 2ql (for M = 1) (4-3)
id V pr
In this case, Ieqd = Ipr. The photocurrent I at the bias voltage of V^ = 30 V; therefore, we can assume Ipr =
I .. with little error. The constant Ie can now be decided by
a vb=30V
29


83
= x ^8,000 x T)2 (8-5)
Very good agreement is obtained as shown in this figure.
The value of and 1/18,000 x T (shot noise level) are
listed in Table 8-3. It shows clearly that at this counting rate,
18,000 counts per minute, for T longer than 100 minutes, 1/f noise
becomes noticeable. For T longer than 1,000 minutes, 1/f noise totally
dominates the noise spectrum.
Besides the fluctuation of scattering cross section described
in Handel's theory, two other possibilities may contribute to the
fluctuation in the total number of counts. First, the pulse height,
which is produced when an a-particle is absorbed by the detector, is a
strong function of the bias voltage applied to the detector [17]. If
the bias voltage was unstable, then a-particles with the same energy
would have been registered in different energy channels. Fortunately,
the detector control unit used in these experiments, Ortec 210, has a
very good stability: bias voltage variation with line voltage is
<0.005% for 105-125 V AC input, and the stability is 0.01% [18]. Secondly,
since a fixed portion of the full spectrum is counted, whether particles
with energies around boundaries fall in or out of boundaries contributes
fluctuations to the total number of counts.
Therefore, a wider energy range should have less fluctuations
in the total number of counts in a certain time period. This is indeed
the case; see Table 8-4. However, for T = 100 minutes, the difference
between R(T)'s for wider range and narrower range is negligible. For


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Associate Professor of Nuclear
Engineering Science
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
P. H. Handel
Professor of Physics
University of Missouri
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August 1983
. /$
Dean, College of Engineering
Dean for Graduate Studies and Research


N
Figure 7-2 R(T) vs. N vor T = 1 min


Table A-7.
Experimental
data for m
0
18,180
T
N

var.
var

0%
Rexp
200
40
3,636,682
5,835,231
1.61
5,036,452
3.808 x 10'7
500
30
9,093,104
18,423,928
2.03
10,914,004
1.320 x 10"7
1,000
15
18,186,209
60,223,202
3.31
26,598,254
8.042 x 10"8
2,000
7
36,369,629
92,776,209
2.55
83,318,101
6.299 x 108
3,000
5
54,558,626
338,426,463
6.20
174,587,904
5.865 x 10"8


Table 8-2. The values of ReXn(T) ancl Rn(T) for T =
ses being the number of measurements
Time (min.)
m
0
100
200
Rpvn(T)
4.126
3.281
27,770/min.
exp
R(T)
6.075(18)
4.255(9)
Rovn(T)
6.359
4.093
17,980/min.
exp
R (T)
nv
6.348(72)
4.087(36)
R n(T)
7.398
3.505
18,160/min.
exp
R(T)
7.437(310)
3.524(155)
R.yn(T)
7.192
8.390
17,180/min.
exp
Rn
6.927(26)
8.257(13)
Rpvn(T)
6.003
1.970
18,356/min.
exp
Rn 6.111(30)
2.024(15)
100-3,000, all values x 10-^, the numbers in parenthe-
500
1,000
2,000
6.105
6.494(3)
3.404
6.8300
3.401(14)
6.8290(7)
0.9662
0.8133
1.3830
0.9742(62)
0.8175(31)
1.3850(15)
8.9990
8.9460(5)
2.3430
4.4280
2.3640(6)
4.4390(3)
3,000
3.360
2.3610(10)


In Equation (5-1) the frequency-shifted components present in
the integral interfere with the elastic term, yielding beats of fre
quency e/tr. The particle density given by Equation (5-1) is
kT|2 = |a2l
1 + 2
j
rA
Iby(e)
cos
f '
+ y
tr e
£0
J
A
b*(e)b-j.(c' Je1' e^^dede'
'0 &0
(5-2)
the second term in large parentheses describes the particle density
beats.
If the particle concentration fluctuation is defined by
2 2 2
6 |xp| = |^| <|ijj| >, its autocorrelation function will be
<<5Ht5Mt+x> = 2
2 a
'A
|b(e)( cos
ex
de
(5-3)
2
which is proportional to |b(e)| and hence proportional to 1/f. There
fore, the spectral density of the particle concentration fluctuation
[the Fourier transform of Equation (5-3)] is proportional to 1/f.
The relative bremsstrahlung rate |b(e)| can be derived as
follows. The constant spectral energy density can be written as
22 3
oo(f) = 4e (AV) /3c k, where e is electronic charge, c is the velocity of
light, < is the dielectric constant of the medium, and AV is the


(n A)
Figure 4-
550
500
450
400
Reach Through Point
1
100
200
I vs. V, for I = 412 nA
a b pr
GO
L
300
j
vb( v )


NPUT
+20V
Figure 3-5. Circuit for a low-noise amplifier (LNA)


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
ABSTRACT v
CHAPTER I: INTRODUCTION 1
PART A
NOISE FROM HOLE-INITIATED
PHOTO AVALANCHE PROCESSES
CHAPTER II: THEORY OF NOISE IN AVALANCHE DIODES 5
CHAPTER III: DEVICE DESCRIPTION AND EXPERIMENTAL SET-UP
FOR MEASURING PHOTODIODE NOISE 20
CHAPTER IV: RESULTS FOR HOLE-INITIATED AVALANCHE CURRENTS 29
PART B
EMISSION STATISTICS OF ct-PARTICLES
CHAPTER V: THEORETICAL PERSPECTIVES 49
5.1 Handel's Theory 49
5.2 The Allan Variance Theorem 52
5.3 Application of the Allan Variance
Theorem to Counting Statistics 54
CHAPTER VI: EXPERIMENTAL SET-UP FOR LONG-TIME COUNTING 57
CHAPTER VII: COUNTING RESULTS 62
7.1 Convergence of the Allan Variance
for N-> 62
7.2 Results of the Relative Allan
Variance vs. 1/T 62
7.3 Results of the Allan Variance vs. T .... 66
CHAPTER VIII: DISCUSSION OF RESULTS 76
i i i


Table A-5. Experimental data for mQ 18,356
T
N

var
var

A2
aMT
R (T)
exp
100
30
1,835,679
3,795,975
2.07
2,022,965
6.003 x 10"7
200
15
3,671,362
10,830,882
2.95
2,654,939
1.970 x 10'7
500
6
9,178,393
64,397,080
7.02
19,739,443
2.343 x 10-7
1,000
3
18,356,785
295,307,233
16.09
149,225,472
4.428 x 10"7
cn


68
Figure 7-5.
R (T) vs. 1/T for m = 17,980/min.
6Xp O


38
which is very close to what we found for previous measurements,
3,733 nA, at a different light intensity. This is in accord with our
assumptions that S, is a constant background noise and independent
of SId.
By comparing the curve of log (Ieq ) versus log (1^) with the
theoretical plots, we note that the curves coincide if the origins
are displaced under 45 by an amount log Ipr, with Ipr = 180 nA, see
Figure 4-6. Figures 4-7 to 4-9 show the results of I. versus V^,
Ieqd versus V^, and Ieqd versus Ij.
Dividing now both axes of Figure 4-5 by the primary current,
we can make a plot of the experimental values
(4-6)
versus
This result is given in Figure 4-10.
The breakpoint structure is not well pronounced, which is due
to the fact that the current gain, or multiplication factor M, for hole-
initiated avalanche current is too small. Therefore, large-voltage
increments are necessary in order to observe current increase while
increasing the bias voltage, and the breakpoint structure could be
easily glossed over. However, we compare the results of Figure 4-10
2
with the theoretical curves of M + var X versus M. The best fit, with
respect to magnitude, is obtained for k = 0.014. The experimental
data plus the theoretical curve is given in Figure 4-11. Comparison of
these experimental results with the asymptotic theory (McIntyre's)
would result in a k value of 0.03, which is too large to believe for a
modern device.


55
Table 5-1. Various
results of Allan
vari anee
theorem
Sm(>
Mt(T>
Poissonian
2mo
= mQT
shot noise
General
shot noise
2 = Km T
T o
1/f noise
2ttC / | co |
2CT2 In 2
Lorentzian
flicker noise
4B a
4b 2 2
B
2
[4e~aT e'2aT + 2aT 3]
a + a)
a
Pathological
noise
L/M^
0 LTX(1 2X"2)
sin (iTA/2)r(A + 1)


CHAPTER I
INTRODUCTION
Avalanche multiplication in devices occurs when the free
carriers gain enough energy in an electric field so that they can
ionize bound carriers upon impact. This phenomenon has been observed
in a number of devices, such as reverse biased p-n junctions, impact
diodes, Read diodes, FET's, etc. Most authors consider the ionization
process as a continuous process [1-6], which means that either the
region over which the avalanching occurs is very long, or the number
of ionizing collisions per primary carrier transit N is infinite or at
least very large.
Ionizations can occur due to electron or hole impact. The
primary carriers responsible for the ionization can be either thermally
generated, or arise from tunneling (as in Zener diodes), or stem from
light absorption; this is the case in avalanche photodiodes, as con
sidered in these studies.
For present-day small dimension devices, the number of ioniza
tions per carrier transit N is small. In 1978 van VIiet and Rucker
used a new statistical approach, referred to as the "method of recurrent
generation functions," to develop a new theory which is valid for an
arbitrary number of N [7,8]. This theory has been corroborated for
electron-initiated ionization process [9].
Measurements with hole-initiated avalanche current are made and
investigated; the resul ts wi 11 be discussed in Part A of this dissertation.
1


31
Vb=30V
Figure 4-1. Id/Id
vs. Vb for Ipr 412 nA


1000
*D
O
V
100
/
/
/
/
/
/
/
/
/
/
/
/
/
101/ 45*
J L
I I 1
-L..-L 1111
10
100
1000
I j ( n A)
Figure 4-2. Log(Ieq^) vs. log(Id) for Ipr
412 nA


54
S
m
2ni
i 00+3
i oo+3
C
dP
eos 2 Ptt
1 2
P-3
r(P)
dT A2m
P aM
OT "t
(5-13)
For Poissonian shot noise S (w) = 2mo, where mQ is the average counting
a?
rate. Substituting Sm(o)) into Equation (5-12), one has 0^(1) = mQT.
2ttC
For 1/f noise, with a spectrum of Sm(w) = where C is a constant,
A2 2
the Allan variance cty| = 2CT £n 2. The various results are summarized
in Table 5-1.
5.3 Application of the Allan Variance Theorem
to Counting Statistics
The presence of 1/f noise in counting statistics can now be
determined from a measurement of the Allan variance as a function of T.
For suppose that the noise is composed of shot noise and 1/f noise, i.e.,
aJ2(T) = mQT + 2CT2 £n 2 (5-14)
recall that = m T, then a measurement of the relative Allan variance
T o
R(T) = a^2(T)/2 yields
R(T) = + 2C £n 2 (5-15)
o
2
where C = C/m is a constant. For short-time intervals the term
o
1/m T is dominant, hence R(T) is proportional to 1/T. When T is long
enough, 2C £n 2 becomes dominant; R(T) is, therefore, a constant,
called the "flicker floor."


CHAPTER VII
COUNTING RESULTS
7.1 Convergence of the Allan Variance for N ->
Because the existence of a "variance of the variance" (or, a
noise of the noise), the relative Allan variance itself is a fluctuat
ing parameter. In order to obtain an accurate value of R(T), a suf
ficient number of measurements must be made especially when T is short.
To determine the minimum number of measurements needed, we plot R(T)
versus N for different T's. These figures show that when N is small,
R(T) is spread over a wide range. When T is increased the spread in
R(T) diminishes and finally R(T) converts to a stable value. For
example, Figure 7-1 shows that for T = 100 minutes, 21 measurements of
M(100) yield an R(100) = (7.09 3.94) x 107; for N = 24, R(T) =
(7.23 2.14) x 107; for N = 27, R(T) = (7.13 1.85) x 10-7. Here
R(T) is given in the form of (mean value standard deviation). From
Figure 7-2 we know that for T = 1 minute at least 70 measurements are
necessary for a reliable value of R(T). Figure 7-3 shows that for
T = 3 minutes we need N ^ 50.
7.2 Results of the Relative Allan Variance versus 1/T
From Equation (5-15) we have
R(T) = + 2C' £n 2 (7-1)
62


12
Converting Equation (2-3) to terms of equivalent saturated
diode current, we obtain
ST = M2St + 2ql var X = 2ql (2-12)
!d Ipr Pr ePd
When the primary current shows full shot noise, we have
Ian /I = M2 + var X (2-13)
eqd/ pr
while
W M f2-14)
Thus, suppose we plot theoretically log (M + var X) versus log (M).
Such a curve is identical to a plot of log (Ieqd) versus log (Id); the
origins of the two curves are displaced by a line under 45, the X-axis
and Y-axis displacements being log (I ). Therefore, by comparing the
experimental curve of Ieq^ versus Id on log-log paper with the theoreti-
2
cal curves for M + var X versus M on log-log paper, we find immediately
Ipr from the displacement of the origin. A special favorable feature
2
is that the theoretical curves of M + var X on log-log paper are very
close for different values of k. Hence k needs only be approximately
known for the determination of I Once I has been found, we then
pr pr
plot Ieq/Ipr vs. Id/Ipr on linear paper, and compare it with curves for
2
M + var X versus M on linear paper. From the magnitude as well as
from the breakpoints in the curve, k is obtained for the best fit. The


71
Figure 7-8. Rexp(T) vs. 1/T for mQ = 18,356/min.


dxe
Poisson Noise
0 Exp. Result
Figure 7-9. R (T) vs. 1/T for m 17,807/min.
GXp U


va r X +M'
19
k : 0.0 2
Figure 2-7.
Var X + M vs,
M for k = 0.02 in linear scale


17
Figure 2-5. Var X + M2 vs. M for k = 0.014 in linear scale


CHAPTER III
DEVICE DESCRIPTION AND EXPERIMENTAL SETUP
FOR MEASURING PHOTO DIODE NOISE
Two specially designed RCA "reach-through" diodes are used
in these hole-initiated avalanche process studies. Figure 3-1 shows the
diode structure and impurity concentration profile. This structure is
designed such that the depletion region in the p-type material reaches
through to the lightly doped it-region when the reverse-bias voltage is
about 200 volts. Further increase in the reverse-bias voltage quickly
depletes the tt-region.
A window is opened on the n+p junction side of the diode so that
light can be shone on the n+ region and then generate electron-hole
pairs. Because the diode is reverse biased, holes will drift into the
high-field region and initiate avalanche processes. The wavelength of
the light is chosen at 502.8 nm so that the absorption depth is shallow;
no light can penetrate to the p-type region, which excludes the possi
bility of electron-initiated avalanche multiplication.
The block diagram of the whole system is shown in Figure 3-2.
The light source is a Spectra Physics 171-08 Argon-ion Laser. The
TEM-01 mode is used. This laser gives a doughnut-shaped light which is
focused by a plain-convex lens. Figure 3-3 shows the biasing circuit
of the device under test (DUT). High Q capacitors are used, so as to
reduce the thermal noise as much as possible. Six 70-volt batteries and
20


Table A-6. Experimental data for mQ 17,807
T
N

var
var

A2
aMT
WT>
50
56
890,370
1,298,893
1.46
963,275
1.123 x 10"6
100
14
1,780,739
3,446,788
1.94
2,583,343
5.616 x 10"7
200
14
3,561,478
8,551 ,758
2.40
5,407,350
2.808 x 10'7
500
5
8,902,655
25,397,533
2.85
18,731,008
1.123 x 10'7


14
Figure 2-2.
Var X + M
2
for k = 0.014 in log-log scale


Table A-4. Experimental data for mQ 17,180
T
N

var
var

A2
aMj
R (T)
exp
100
26
1,717,908
10,533,131
6.13
2,122,404
7.192 x 107
200
13
3,435,815
38,770,910
11.28
9,904,640
8.390 x 10'7
-7
500
5
8,589,200
186,528,449
21.72
66,392,064
8.999 x 10
UD
4^>


NORMALIZED ALLAN VARIANCE
A?
Figure 8-2. vs. T for mQ = 18,000/min.


3.4X1015 -*2/4Ln LD=1.7Hm
A '
n =
^1/2 I
TT L,
P =
5 X 1 012
TT 1 7 2 L
TT La
-V1
*"A5 7.6 pr
p"^ = 1 m deep
p m
Figure 3-1.
Diode structure and impurity concentration profile


(nA)
35
-o
O-
4000
3000
2000
1000
0
vb(v)
0 100 200 300 400
ec>d
Figure 4-4. I
vs. for I = 412 nA


5 722
No ¡se Diode
Noise Output
Figure 3-4. Circuit of a noise standard using a vacuum diode


HO
1 1-1 1.2 1.3
M
Figure 4-10.
Var X + M vs. M for Ipr = 412 nA


=0.016
Figure 2-1.
y vs. M


PART B
EMISSION STATISTICS OF a-PARTICLES


47
Figure 4-13.
Var X + M
2
vs. M with theoretical curve for I
pr
180 nA


pba
41
Figure 4-8. I vs. V, for I = 180 nA
eqd b pr


exp
Poisson Noise
-4
10
O Exp. Result
-5
10

a
q/3
7

ac
/
-6
10
: /
-7
10
7
0 /
r
1 1 1 L-
-3 -2 -1
10 10 10 i
(min.-1 )
Figure 7-6.
R (T) vs. 1/T for m = 18,160/min.
c A [J U


5 pF-200 pF
70V-
70V-
70 V-
70 V-
70 V-
70V-
Figure 3-3. Biasing circuit of the DUT
h-KkKhl
20 V
O.I/iF 0.1/i.F 100ft J
7mH c

5.5M)
6)
(oj
NOISE
OUTPUT
>AV7
30M1
lV\AA/
IK
O.I/xF
2N 4220A
TO
LNA
IV
o


Figure 6-1. Block diagram counting system


45
2
The experimental M + var X versus M curve obtained from lower
light intensity measurements is presented in Figure 4-12. Comparison
with the theoretical curve for k = 0.014 is made in Figure 4-13. This
diode shows consistent noise performance while operated at different
light intensities.
From the data presented above for low M values and various
light intensities, we conclude that Van Vliet and Rucker's theory is
applicable for hole-initiated avalanche currents.


where C is a constant, mQ is the averaged counting rate, and T is
the counting time.
Figures 7-4 to 7-10 show seven series of experimental results
of R(T) versus 1/T. Figures 7-4 to 7-6 show clearly that for T small
(1/T large), R(T) is equal to the value of Poissonian shot noise, the
first term of Equation 7-1. But for T large (1/T small), the values
of R(T) is larger than the value for shot noise. Figures 7-7 to 7-10
show the results of some experiments which were concentrated on long
time measurements. They all show that for T long enough the experimental
values of R(T) are higher than l/mQT and the curves start to level off,
which implies the existence of 1/f noise. The second, constant term in
Equation (7-1) becomes dominant.
7.3 Results of the Allan Variance versus T
A 2
Figures 7-11 and 7-12 give the plot of the Allan variance, aM ,
versus T corresponding to the measurements of Figures 7-4 and 7-5,
A2
respectively. Recall that for shot noise = mQT, which is propor-
A2 2
tional to T, but for 1/f noise qv|y = 2CT £n 2, which is proportional
2
to r.
Since these figures (Figures 7-11 and 7-12) are plotted on
log-log scale, the slope of log (o$y) versus log(T) is really the
A2
exponent of T in the expression of a^y, see Equation (5-14). Figures
A2 2
7-11 and 7-12 show clearly the dependence of c^y on T and T while
different noise terms dominate the noise spectrum.


¡qd (n A )
3500
3000
2500
2000
1500
1000
500
200 1
400
450
500
j
Ij (nA)
pr
412 nA
Figure 4-5. 1^ vs. Id for I


Immediately following the LNA is the HP-31OA Wave Analyzer,
which is used essentially as a narrow-band, high-gain, tunable filter.
It is used in the AM mode with a bandwidth of 3 KHz.
The output of the wave analyzer is connected, through an RC
low-pass filter (or an integrator), to the Y-axis input of the MFE 815
X-Y plotter with an MFE 7T time base on the plotter's X-axis input
terminals. The Y-direction displacement is proportional to the output
voltage of the HP-310 Wave Analyzer and its time average over a certain
period of time can be obtained from this plotter. This time-averaged
voltage reading gives more accurate results than that obtained directly
from an rms volt meter.
The whole system is first operated with both the light and
calibration noise source off. The variable capacitor (see Figure 3-3)
is tuned at the minimum capacitance position and the receiver (HP-310
Wave Analyzer) is set to the resonant frequency of the biasing circuit,
203 KHz, such that the maximum noise output is obtained. A voltage
meter reading M-j is recorded and the noise power at the output of the
aave analyzer in this case is
M
2
1
(3-1)
where B is the bandwidth of the whole system, G the power gain of the
amplifier, and stands for the spectral density of background
noise, which includes the amplifiers noise and noise due to the dark
current of the diode.
Next, we turn the calibration noise source on and record the
meter reading M^ and the average anode current of the 5722 saturated


times; hence, both ADCs covered exactly the same time span. The "add-
up" method was applied to M-jqqS, obtained from ADC #1, to find out the
calculated M^qS.
Table 6-1 lists the results calculated from both ADCs for
500-minute measurements. The difference between them, in each category,
is <1%. This shows the validity of the "add-up" method.
Table 6-1. Comparison of the results obtained
from the "add-up" method (ADC #1)
and the real-time measurements
(ADC #2)
ADC #1
ADC #2
A
o
o
LO
9,062,464.36
9,067,009.84
A2
0^5OO
7,862,171.3
7,943,285.5
R(500)
9.573 x 10-8
9.662 x 10'8


CHAPTER IX
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
From the experimental data presented in the Appendix, we see
that the ratio for the averaged value of the total number of counts,
, to the variance of M is very close to one when T is less than
100 minutes, which is an indication for Poissonian statistics. However,
when T is greater than 100 minutes this ratio becomes greater than
unity and generally increases monotonically with T, which indicates
super-Poisonian statistics. With the help of the newly devised Allan
variance theorem, we found that this excess noise in the radioactive
decay rate has a 1/f spectrum.
Quantitatively, from Equation (5-18), we know the flicker floor
F has a value of
F = 126.4 x 10'7 x (9-1)
K
To the author's knowledge, nobody has ever measured the dielectric
constant of Am02, the a-particle source used in these experiments; thus
the value of k in Equation (9-1) is unknown. However, since the atomic
structure of Am02 is similar to that for U02, the dielectric constant
of U02 (20.4 1.5 [19], 21.7 0.5 [20], and 21.0 1 [21]) can then
be used as a reference.
If one considers the following factors that (1) the coherence
factor, C, is always less than unity; (2) the dielectric constant of
87


74
A2
Figure 7-11. aM (T) vs. T for rn = 27,700/min.
' 0


50
w(
Figure 5-
Spectral density of bremsstrahlung in power W and
photon emission rate N


CHAPTER V
THEORETICAL PERSPECTIVES
5.1 Handel1s Theory
It is known that upon scattering a beam of electrons will
emit bremsstrahlung. The power spectrum W(f) of the emitted radiation
is independent of frequency (W = constant) at low frequencies and
decreases to zero at an upper frequency limit f which is approximated
by E/h, where E is the kinetic energy of the electrons, h the Planck
constant. Consequently, the rate of photon emission per unit frequency
interval is N(f) = W/H^, i.e., proportional to 1/f (see Figure 5-1).
Therefore, we conclude that the fraction of electrons scattered with
energy loss e is proportional to 1/e: i.e., the relative squared matrix
element for scattering with energy loss e is [by(e)[ ^ 1/e.
If the incoming beam of electrons is described by a wave func
tion exp[(i/TT) (F. r Et)], the scattered beam will contain a large
nonbremsstrahlung part of amplitude a, and an incoherent mixture of
waves of amplitude aby(e) with bremsstrahlung energy loss e ranging
from some resolution threshold eQ to an upper limit A < E, of the order
of the kinetic energy E of the electrons,
^T
*
d
exp
.TTj
(P r Et)
a
1 +
4
byieje"* £^'/^'de
(5-1)
I I
where by(e) = |by(e)|e has a random phase Y£ which implies incoherence of
2
all bremsstrahlung parts, and |b-j-(e) | is proportional to 1/e.
49


88
Am02 may be larger than 21, then the measured value of F = 1 x 10
is in the right ballpark to verify Handel's theory. In particular,
for c = 0.17, k = 21, one obtains F = 1.02 x 10'., in accord with the
observed value of Figures 8-1 and 8-2. We believe, therefore, that
these experiments constitute the first experimental verification of
Handel's quantum 1/f noise theory.
241
The data presented here all referred to g^Am If an a-source
with different energy peaks is used, the counting could then be done
in each of these energy peaks. This should result in an R(T) versus 1/T
as given in Figure 9-1.
If Handel's theory is correct, the flicker floor should be
2 2
proportional to v /c i.e., to the energy E. Thus, the flicker floors
shown in Figure 9-1 should have the ratios of these energy peaks. This
experiment will therefore be a very strong test to verify the appli
cability and correctness of Handel's theory.


Table A-3. Experimental data for mQ 18,160
T
N

var
ya.r.

A2
aMj
R (T)
exp
1
248
18,160
16,829
0.93
16,647
5.048 x 10-5
2
124
36,321
34,706
0.96
34,714
2.631 x 10'5
5
49
90,795
64,814
0.71
59,228
7.185 x 10"6
10
88
181,684
214,732
1.18
217,969
6.603 x 10'6
20
44
363,367
436,054
1.20
450,422
3.411 x 10-6
50
149
908,647
1 ,274,359
1.40
936,880
1.135 x 106
100
310
1,812,493
9,116,358
5.03
2,430,287
7.398 x 107
200
155
3,625,004
31 ,272,439
8.63
4,606,285
3.505 x 10"7
500
62
9,067,010
168,503,103
16.58
7,943,286
9.662 x 10'8
1,000
31
18,134,020
668,501 ,571
36.86
26,745,242
8.133 x 10-8
2,000
15
36,265,809
2,626,291,950
72.42
181,946,661
1.383 x 10-7
3,000
10
54,398,714
5,991 ,195,130
110.14
698,293,362
2.360 x IQ*7


9
respectively. See Figure 2-1. At the M value for which the switch
over occurs, the variance jumps from var X-j to var 1^- Equation (2-10)
indicates that the decrease in y and the increase in N cause var X
to make a positive jump at this particular M, which we call a break
point value.
If now the field is further increased, the paths £-j +
decreases, y(2) increases, and M increases, until at y = y(2) three
TlaX
ionizations per carrier transit are possible. Once the regime N = 3 is
possible, the y(3) for these processes is much smaller than y(2) in
max
order to realize the same M. The switch-over marks another breakpoint
value for M, at which the variance jumps due to the reduction of y, from
y(2)max to y(3), etc. We thus obtain an overall curve of var X versus
M, which shows discontinuities at the breakpoint values of M, for which
the regime switches N -> N + 1. In order to construct this overall
curve we need only plot Equation (2-10) with y expressed as y(M,N)
according to the inversion of (2-9), where N = 1 and y = y(l) up to
y(1 )max next N = 2 and Ia = v(2) up to u(2)max> etc.
We conjecture that y(N)max is related to the pole value,
y (N), for which the denominator of (2-9) is zero. These pole values
r
are listed in Table 2-1. On general ground we assumed that the ymgx is
related to yp by
y(N)
N
max N + 5 Mp
yn(N)
(2-11)
The reason for this presumed relationship is that for very large
N we must have y(N) --y (N) in order that M can rise without saturation.
K max 'p


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NOISE ASSOCIATED WITH ELECTRON STATISTICS
IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF a-PARTICLES
By
Jeng Gong
August 1983
Chairperson: C. M. Van Vliet
Major Department: Electrical Engineering
It is known that the standard theories of avalanche statistics
for two-carrier impact ionization by Tager, McIntyre, and Personick
only deal with processes for which the number of possible ionizations
is very large. On the contrary, it is believed that in many modern
devices the number N of possible ionizations is finite and perhaps even
very smal1 (N = 1 5).
In 1978, van Vliet and Rucker developed a new theory for-this
small N case, which has been fully confirmed for the electron-initiated
ionization process. Part A of this dissertation presents the results
of a detailed noise study with hole-initiated avalanche currents which
corroborate van Vliet and Rucker's theory for the second time.
Data obtained from extensive measurements on counting techniques
241
for a-particles radioactive decay from g^Am are presented in Part B
v


OIxcoa
Figure 7-3. R(T) vs. N for T = 3 min.


Table 4-2. Experimental data for I = 180 nA
V
^eqj^eq 3,782)
Id/180
*eq(j/T80
30
180
3,962
180
1.000
1.00
100
181
4,039
257
1.006
1.43
135
185
4,050
268
1.028
1.49
150
190
4,104
322
1.056
1.79
175
195
4,172
390
1.083
2.17
200
199
4,234
452
1.106
2.51
240
204
4,309
527
1.133
2.93
260
208
4,407
625
1.156
3.47
280
214
4,506
724
1.189
4.02
300
220
4,608
826
1.222
4.59
315
226
4,793
1,011
1.256
5.62
330
231
4,967
1,185
1.283
6.58
340
237
5,217
1,435
1.317
7.97
350
244
5,526
1,744
1.356
9.69


I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
/vV l/isu l
Carolyn M. Van Vliet, Chairperson
Professor of Electrical Engineering
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
C/$
A. van der Ziel ^
Graduate Research Professor of
Electrical Engineering
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
A. D, Sutherland
Professor of Electrical Engineering
I have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
R_ cA^
E. R. Chenette
Professor of Electrical Engineering


(4-4)
For the data presented in Table 4-!, IeqA 3 ,733 nA; Igq^ can then be
found by using Equation (4-2).
After obtaining Ieqd> we plot log (Ieqd) versus log (Id). As
discussed in Chapter II, the primary current can then be decided by
comparing this plot with theoretical curves. The Ipr so determined in
Figure 4-2 is exactly equal to what has been assumed, namely Ipr =
The results of Ieq W^pr5 and leq^pr are a^so 9iven
in Table 4-1.
In the data given here, we show the following types of measure
ments: (1) the avalanche photocurrent Id as a function of bias voltage;
(2) the noise, as expressed in the equivalent saturated diode current
Ieqd, as a function of bias voltage; (3) Ieqd as a function of diode
current Id. The measurements are typical of many others. We determined
the diode current Id and the noise Ieqd for different light intensities.
The data for different light intensities are quite similar and generally
consistent with each other as predicted by the theory.
Figures 4-3 to 4-5 show the experimental data obtained for a
primary current Ipr of 412 nA. The Id vs. Vd graph shows a slight bend
which marks the reach-through voltage. The noise is plotted in Figures
4-4 and 4-5 as a function of and Id, respectively.
Table 4-2 lists the data obtained for a different light inten
sity. The Ieq^ for these measurements is obtained by Equation (4-4).
I = 3,962 nA 180 nA = 3,782 nA
eq£
(4-5)


16
Figure 2-4.
Var X + M2
vs. M for k = 0.02 in log-log scale


30
Table 4-1. Experimental data for I = 412 nA
V
Id
!eq
I-eq(-j(I-eq 3,733)
'd/412
W412
30
412
4,145
412
1.000
1.000
100
417
4,159
426
1.012
1.030
150
422
4,180
447
1.024
1.085
170
432
4,268
535
1.049
1.299
185
442
4,480
747
1.073
1.810
200
449
4,637
904
1.090
2.190
240
458
4,722
989
1.112
2.400
260
469
4,792
1,059
1.138
2.570
275
480
5,058
1,325
1.165
3.220
300
492
5,352
1,619
1.194
3.930
320
512
5,887
2,154
1.243
5.230
340
532
6,855
3,122
1.291
7.580


exp
70
Figure
7-7. R (T) vs. 1/T for m = 17,180/min.
exp o


CHAPTER VI
EXPERIMENTAL SET-UP FOR LONG-TIME COUNTING
The block diagram of the counting system being used to investi
gate 1/f fluctuations in the a-particle emission rate is shown in
241
Figure 6-1. The source is g^Am which decays with a half-life time
of T-j/2 = 458 years with the emission of 5.48 MeV a-particles into
237
g2Np The detector, a silicon surface-barrier detector, is reverse
biased at 80 volts, and the dead times of the ND575 Analog-to-Digital
Converter (ADC) and ND66 Multi-Channel Analyzer are 60 n-seconds and
6 y-seconds, respectively. Therefore, no dead-time correction is neces
sary, as long as the counting rate is kept lower than 1,000 counts per
second [16] (or the averaged time elapse between two counts is higher
than 1,000 y-seconds).
A typical full-energy spectrum measured in these experiments
is shown in Figure 6-2 in semi-logarithmic scale. The spectrum is
shown on a display screen while accumulating counts and the final
result, after a chosen time T, is stored in the memory units of the ND66.
The Full Width Half Maximum (FWHF) of the spectrum can be found by
moving the cursor on the display, which indicates the number of counts
under each energy channel. The systematic range used in these experi
ments was from peak channel -FWHM x 6 to peak channel +FWHM x 2. There
fore, the total number of counts M-j., which will be analyzed later, is
always under a fixed portion of the full spectrum.
57


It is not necessary to commit oneself to Poisson statistics
of the primary particles; so, one can consider immediately the
statistics of the avalanche process. Thus let X denote the offspring
(plus original carrier) due to one incoming primary carrier. Then,
according to the variance theorem,
Sj = M2Si + 2qlpr var X (2-3)
where M = . From this we find that (2-1) and (2-2) are tantamount
to
var X = M^(M 1) (Tager)
(2-4)
and
var X = M(M 1) + k(M 1)^M (McIntyre) (2-5)
The above theories assumed that the region over which avalanche
occurs is very long, so that the number N of possible ionizations per
carrier transmit is very large. Therefore, these theories can be
referred to as asymptotic theories. Lukaszek et al. at the University of
Florida (1974) were the first authors to show that this assumption is
unrealistic [14], in particular for the region of onset (low gain) of
avalanche multiplication. They developed a theory for the case that N = 1
and N = 2. Besides for short avalanche regions, these formulas should
always be applicable at the onset of avalanche ionization. For N = 1,
their results give


Page
CHAPTER IX: CONCLUSIONS AND RECOMMENDATIONS FOR
FUTURE WORK 87
APPENDIX: EXPERIMENTAL DATA FOR COUNTING EXPERIMENTS .... 91
REFERENCES 98
BIOGRAPHICAL SKETCH TOO
IV


of this dissertation. These data have shown that the statistics are
non-Poissonian for large counting times (order 1,000 minutes) in
contrast with the fact that many textbooks cite a-decay as an example
for Poisson statistics.
Detailed measurements of the Allan variance indicated a
"flicker floor" due to the presence of 1/f noise in the decay, of
10"7. This result is in agreement with Handel's quantum 1/f noise
theory. If upheld by further measurements, then this would be the
first quantitative indication that 1/f noise is caused by emission of
long wavelength infraquanta, such as soft photons causing minute
inelastic losses in the scattered wave packet.
vi


X + M
44
Figure 4-11.
Var X + M2
vs. M with theoretical
curve for I = 412 nA
pr


NOISE ASSOCIATED WITH ELECTRON STATISTICS
IN AVALANCHE PHOTODIODES
AND EMISSION STATISTICS OF a-PARTICLES
By
JENG GONG
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA


REFERENCES
[1] A. S. Tager, "Current fluctuations in a semiconductor (dielec
tric) under the conditions of impact ionization and avalanche
breakdown," Sov. Phys.Solid State, Vol. 6, pp. 1919-1925, 1965.
[2] R. J. McIntyre, "Multiplication noise in uniform avalance diodes,"
IEEE Trans. Electron Devices, Vol. ED-13, pp. 164-168, 1966.
[3] R. J. McIntyre, "The distribution of gains in uniformly multi
plying avalanche photodiodes: Theory," IEEE Trans. Electron
Devices, Vol. ED-19, pp. 167-190, 1971.
[4] S. D. Personick, "New results on avalanche multiplication statis
tics with applications to optical detection," Bell Syst. Tech. J.,
Vol. 50, pp. 167-190, 1971.
[5] S. D. Personick, "Statistics of a general class of avalanche
detectors with applications to optical communication," Bell Syst.
Tech. J., Vol. 50, pp. 3075-3094, 1971.
[6] G. E. Stilman and C. M. Wolfe, "Avalanche photodiodes," in
Semiconductors and Semimetals, Vol. 12. R. K. Willardson and
A. C. Baer, Eds. New York: Academic Press, 1979, Ch. 5.
[7] K. M. van Vliet and L. M. Rucker, "Theory of carrier multiplication
and noise in avalanche devicesPart I: One-carrier processes,"
IEEE Trans. Electron Devices, Vol. ED-26, pp. 746-751, 1979.
[8] K. M. van Vliet and L. M. Rucker, "Theory of carrier multiplication
and noise in avalanche devicesPart II: Two-carrier processes,"
IEEE Trans. Electron Devices, Vol. ED-26, pp. 752-764, 1979.
[9] J. Gong, K. M. Van Vliet, A. D. Sutherland, and E. R. Chenette,
"Noise measurements on photo avalanche diodes," Phys. Stat.
Sol. (a) 63, pp. 445-460, 1981.
[10] P. H. Handel, "Quantum approach to 1/f noise," Physical Review
(A), Vol. 22, No. 2, pp. 745-757, 1980.
[11] K. L. Ngai, "A unified theory of 1/f noise and dielectric
response in condensed matter." Proceedings of second inter
national symposium on 1/f noise, pp. 445-476, 1980.
98.


Table 8-2Continued
Time (min.)
m
0
100
200
R n(T)
8.147
4.263
17,807/min.
exp
Rn
8.087(28)
4.233(14)
R nlD
3.808
18,180/min.
exp
Rn
3.836(40)
500
1,000
2,000
5,000
2.3630
2.3510(5)
1.3200
0.8042
0.6299
0.5865
1.3310(30)
0.8099(15)
0.6327(7)
0.5884(5)


counting rates. Therefore, the shot noise term in the relative Allan
variance, R(T), is slightly different for each series of measure
ments .
Before the average of Rexp(T) is taken, the shot noise term,
¡yy should be normalized to the same counting rate. Here a rate of
o
18,000 counts per minute was chosen. The normalized relative Allan
variance R (T) is then
Rn^ = Rexp^ mV + 18,000 x T = 18,000 x T + 2C' 2
(8-3)
Tables 8-1 and 8-2 show the values of ReXp(T) and Rn(T). Instead
of <'ReXp(i)> is now used to estimate the true value of R(T).
The value of is calculated as follows:
ZR .(T) x DF
= ni 1
ZDF.
(8-4)
f"h
where R .(T) is the value of R (T) obtained from the i series and DF.
m n i
stands for the degrees of freedom of that particular value, which equals
the number of measurements minus one.
Figure 8-1 shows the comparison of and (1/18,000 x T)
+ 1 x 10-^ versus 1/T, which suggests that the value of 2C £n 2 is
about 1 x 10"^.
Figure 8-2 shows the comparison of the average value of the
A2
normalized Allan variance, , which is obtained by


75
T (m in.)
(T)
vs.
T for m =
o
Figure 7-12.
17,960/min.


7
var X = M(M 1) 1 +
IT 1%rrl-pj (Lukaszek) (2-6)
where X is the a priori chance for ionization by an electron after the
electron has gained enough energy to ionize, whereas y pertains to ioni
zation by a hole.
In 1978 van VIiet and Rucker reinvestigated the problem. A new
theoretical method was developed, named the "method of recurrent
generating functions" [7,8], By this method they were able to solve the
complete problem, in which N possible ionizations per carrier transit,
or per traveling hole-electron pair (the electron and hole going in
opposite directions) are possible. In this case the basic parameters
are the a priori ionization probabilities by electron impact (x) or by
hole impact (y), once these carriers have covered a path which is long
enough to gain the necessary ionization energy from the electrical
field.
For the case that the primary carriers injected into
the avalanche region are electrons, the following results are found
(1 + A)N(1 k)
(2-7)
(1 + kX)N+1 k(l + X)N+1
var X e <(AX)2> =
2
2 + X + kX
X
(2-8)


99
[12] S. Machlup and T. Hoshiko, "Scale invariance implies 1/f spec
trum," Proceedings of second international symposium on 1/f
noise, pp. 556-558, 1980.
[13] C. M. Van Vliet, A. van der Ziel, and P. H. Handel, "Super-
statistical emission noise," Physica 108A, pp. 511-526, 1981.
[14] W. Lukaszek, A. van der Ziel, and E. R. Chenette, "Investigation
of the transition from tunneling to impact ionization multipli
cation in silicon p-n junctions," Solid-State Electron., Vol. 19,
pp. 57-71, 1976.
[15] C. M. Van Vliet and P. H. Handel, "A new transform theory for
stochastic processes with special application to counting
statistics," Physica 113A, pp. 261-276, 1982.
[16] G. F. Knoll, "Radiation detection and measurement," New York, Wiley
and Sons, p, 95, 1979.
[17] W. J. Price, "Nuclear radiation detection," New York, McGraw-Hill,
p. 273, 1958.
[18] Ortec technical data, 210 detector control, 1971.
[19] A. D. B. Woods, G. Dolling, and R. A. Cowley, "The crystal
dynamics of uranium dioxide," Inelastic scattering Neutrons,
Proc. Symp., 4th, Bombay, pp. 373-378, 1964.
[20] D. J. Huntley, "The dielectric constant of U02 and its variation
with porosity," Can. J. Phys. 44(11), pp. 2952-2956, 1966.
K. Gesi and J. Tateno, "Dielectric constant of uranium dioxide
at 9.4 GH2," Jap. J. Appl. Phys., 8(11), pp..1358-1359, 1969.
[21]


Table A-l. Experimental data for mQ 27,700
T
N

var.
var

A2
aMT
WT)
1
96
27,761
29,759
1.07
26,874
3.487 x 10'5
2
48
55,525
64,254
1.16
63,300
2.053 x 10"5
5
19
138,797
174,979
1.26
162,334
8.427 x 10"6
10
38
277,369
271,146
0.98
300,625
3.907 x 106
20
18
554,689
407,757
0.74
366,675
1.191 x 106
50
7
1,386,606
1,556,691
0.83
950,741
4.945 x 10"7
100
18
2,772,026
13,322,534
4.81
3,170,123
4.126 x 10"7
200
9
5,544,053
49,414,157
8.91
10,084,511
3.281 x 10-7
500
3
13,855,521
202,833,876
14.64
117,194,752
6.105 x 10~7


(n A)
39
10
100
1000
Id (n A)
Figure 4-6.
Log(1eqd) vs- 19(Id) for TPr
180 nA


Table A-2. Experimental data for mQ 17,980
T
N

var.
var.

A2
aMj
R exp
1
177
17,987
17,155
0.95
18,534
5.729 x 10'5
2
88
35,970
29,055
0.81
30,030
2.321 x 10"5
5
93
89,984
108,003
1.20
121,148
1.496 x 10"5
10
83
185,379
154,777
0.84
141,464
4.116 x 106
20
41
370,752
324,936
0.88
311,757
2.268 x 10'6
50
16
926,836
868,989
0.94
1,000,977
1.165 x 10"6
100
72
1 ,796,155
15,133,266
8.43
2,051,505
6.359 x 107
200
36
3,592,309
55,659,746
15.49
5,282,260
4.093 x 10-7
500
14
8,980,092
324,379,609
36.12
27,448,241
3.404 x 10'7
1,000
7
17,960,183
1,311,457,070
73.02
220,310,187
6.830 x 107


Figure 4-7. 1^ vs. for I
180 nA