NOISE IN PHOTOTRANSISTORS

By

FRANCISCO H. DE LA MONEDA

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

1970

ACKNOW ILE DGMENTS

I wish to express my sincerely felt appreciation to Dr. E. R. Chenette

and Dr. A. van der Ziel for their guidance and enlightening discussions

that have made it possible to carry this investigation to a fruitful con-

clusion. I also wish to thank Dr. K. B. Cook who graciously provided che

information on the transistor-fabrication process.

This investigation was supported by the Advanced Research Projects

Agency, U. S. Department of Defense and monitored by the Air Force

Cambridge Research Laboratories under Contract No. F 19628-6S-C-005S.

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . ..... . . . . . . . . .ii

LIST OF TABLES . . . . . .... .. . . . . . .. v

LIST OF FIGURES . . . . . .... . . . . . . . vi

LIST OF SYMBOLS . . . . . . . . . . ........ v iii

ABSTRACT . . . . .... . ........... . . . xii

CHAPTER

I INTRODUCTION . . . . . .... . . . . . . 1

II ANALYSIS OF D.C. AND A.C. PHOTOTRANSISTOR CHARACTERISTICS . 4

D.C. Characteristics . . .... . . . . . ... 4

Gain Mechanism . . ..... .. . . . * 8

Quantum Efficiency . . . . . . . . . . 10

Definition . . . . . . . . . . .. 10

Analytical solution of the quantum efficiency . .. .10

Numerical solution of the quantum efficiency . . 13

A.C. Characteristics ... . . . . . . . 19

Transistor Small Signal Equivalent Circuit ........ 19

A. C. Quantum Efficiency . . . .... . . . . 19

Responsivity . . . .... . . . . . . 23

III NOISE CHARACTERIZATION .. . . . . . . . . ... 24

Collector-Base Photodiode Noise Current . . .... . . 25

Output Noise Current at Low Frequencies . . . . .. 27

Output Noise Current Including High-Frequency Effects . . 55

NEP, SNEP, and D* . . . . . . . . . 55

IV EXPERIMENTAL RESULTS . . . . . . .... .. . 537

Noise Measurements . . . ..... .. . . * * 357

Introduction . . . . ... . . . * 357

Method of Measurement . . . .... . . . 40

A.C. Measurements . . . .... . . . . . . 45

Transit-Time Measurement . . . .... . . . . 45

Frequency Response . . . . . . . . . 46

D.C. Measurements . . . . . . . . . . .

Noise Spectra . . . . . . . . . . . .

Texas Instrunents Phototransistor Type LS-400 ..

Motorola Phototransistor Type MRD-310 . . . . .

Fairchild Phototransistor Type FPT-100 . . . . .

University of Florida Phototransistor . . . . .

Computation of High-Frequency Effects . . . . . .

V CONCLUSIONS . . . . . . . . . . . . .

APPENDICES

A NUMERICAL SOLUTION OF THE CONTINUITY EQUATION . . . .

B PHOIOTRANSISTOR FABRICATION . . . . . . . . .

Fabrication Steps . .

Wafer Cleaning . .

Oxidation . . .

Base Diffusion . .

Emitter Diffusion .

Cettering of Metal Prec

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

ipitates . . . . . .

BIBLIOGRAPHY ... . . . . . . . . . . . . . .

BIOGRAPHICAL SKETCH . . . . . . . . . . . . .

LIST OF TABLES

Table Page

4.1 Comparison Between the Low- and High-Frequency

Components of the Output Noise . . . . . .... 68

B.1 Physical Parameters of the U. of F. Phototransistor . . 7

B.2 Electrical Parameters of the U. of F. Phototransistor .. 88

LIST OF FIGURES

Figure No. Page

2.1 Cross-sectional view of planar n-p-n phototransistor . . 6

2.2 One-dir-ensional models of phototransistors. a) For

low-injection level. b) For high-injection level . . 6

2.5 Bulk current components for an n-p-n phototransistor.

Carrier sources and sinks are also shown . . . . 7

2.4 Computed and experimental collector-base photodiode

quantum efficiency. . . . . . . . . ... 15

2.5 Computed collector-base photodiode quantum efficiency . 16

2.6 Computed and experimental collector-base photodiode

quantum efficiency . . . . . . ... ... 18

2.7 Computed and experimental collector-base photodiode

quantum efficiency . . . . . . . ... 20

2.8 Transistor equivalent circuit, a) Hybrid-pi model

with collector-base excitation source. b) Equivalent

configuration. . . . . . . . ... .. 21

5.1 Phototransistor noise model. Shot noise current

generators i and ib are imposed on a hybrid-pi small

signal equivalent circuit . . . . . . . 51

4.1 Noise resistance vs. source resistance of preamplifier

used for noise measurements. . . . . . . ... 59

4.2 Noise measurement set-up. . . . . . . . ... 41

4.5 Transit-time bridge used to make small signal measurements

of hfe and fB. ................... .. 44

4.4 Set-up used to measure the phototransistor f using an

amplitude-modulated-light signal from a GaAs diode . 47

4.5 Bridge used to measure hFE............... 47

4.6 Measured and computed I vs. frequency plots for T-1 LS-400

phototransistor .eq ..... 49

Figure No. Page

2

4.7 Plots of h f/hFE and f vs. ICEO for T-I device

using spectra in Figure 4.6 and Eq. [5.19] . . . ... .50

4.8 Measured I vs. frequency at various temperatures for

eq

the T-I device . . . . . . . . ... .... .. 51

4.9a Measured I vs. frequency for Motorola MRD-510

eq

phototransistcr for ICEO up to 10 pA . . . . . . 55

4.9b Measured I vs. frequency for Motorola MRD-510

eq

phototransistor for ICEO up to 1 rA . . . . . . 54

CEO

2

4.10 Plots of m hF mhFE' and hE vs. IE for Motorola device.

Values of m hFE were obtained using the spectra in

Figure 4.9 and Eq. [5.19]. . . . . . . . . ... 55

4.11 Plots of m(IE) computed from curves in Figure 4.10

for Motorola device. . . . . . . . . ... .. . 56

4.12 f vs. ICEO for Motorola device . . . . . . .... 57

4.13 Measured I vs. frequency for Fairchild FPT-100 phototransistor.

eq

I was measured for both transistor and phototransistor

eq

modes of device operation . . . . . . . .... 58

4.14 Comparison between measured and computed I for

Fairchild device . . . . . .eq . . . . . 59

4.15 Measured I vs. frequency for Motorola unit for both

eq

transistor and phototransistor modes of operation . . . 61

4.16 Measured I vs. frequency for University of Florida

eq

phototransistor ....................... 65

4.17 Plots of m hFE, mhFE, and hFE vs. IE for University of

2

Florida device. Values of m hFE were obtained uning the

spectra in Figure 4.16 and Eq. [5.19] ........ . . 64

4.18 Plots of m(IE) computed from curves in Figure 4.17 . . . 65

4.19 f vs..1E for University of Florida device . . . . .. 66

A.1 Hypothesized dependence of the v(XB) on v(0) . . . . . 7

vii

LIST OF SYMBOLS

The following list contains most of the important symbols used in

the text. Following tradition, capital letters denote d.c. quantities and

lower case symbols denote a.c. quantities. This also applies to subscripts

most of the times.

Ab area of the base region exposed to radiation

B Boson factor

C doping impurity concentration

CCT collector-base junction transition capacitance

CET emitter-base junction transition capacitance

c velocity of light

D electron diffusivitv in the base region

n

D spectral detectivity at wavelength X

E electric field in the base region

F(v) photon flux: race per unit area at optical frequency v

f electrical frequency

f beta cut-off frequency

gceo lou-freLuency value of the real part of the collector-

emitter transadmittance

ge real part of the admittance of the emitter-base junction

geo low-frequency value of ge

h Planck's constant

hFE d.c. current gain, Eq. (2.5]

viii

hfe phototransistor incremental current gain

IC d.c. collector current

IBC collector-leakage current due to thermal excitation

ICEO d.c. collector or emitter current

ID coefficient of the electron current injected from

emitter to base

IE d.c. emitter current

IET coefficient of the emitter-junction-leakage current

I equivalent shot noise current corresponding to the

eq phototransistor output noise, Eq. [3.19]

IN photogenerated current in the collector

IND noise diode d.c. current

Ip photogenerated current in the base

IPH photogenerated current

ib instantaneous base current

i instantaneous collector current

c

i instantaneous emitter current

e

i instantaneous short-circuited output current of a

phototransistor

ph small signal photogenerated current flowing in the

collector-base photodiode

is (vs,f) small signal output current of a phototransistor due

to j (vs'f)

J electron current density in the base region

n

J hole current density in the base region

2

j imaginary unit such that j =-1

j(vs,f) small signal photon flux rate per unit area at optical

frequency v and electrical frequency f

s

k Boltzmann's constant

L diffusion length of electrons in the base

n

L diffusion length of holes in the base

P

m(LE) ratio of hfe to hFE

m' reciprocal slope of the emitter-leakage current vs.

forward junction voltage

N number of photons in modes within a unit optical

frequency interval

NEP noise equivalent power

n total free electron concentration in the base

n equilibrium electron concentration in the base

n(x,t) total free electron concentration in the base at a

given depth x and time t

p total free hole concentration in the collector

Pn equilibrium hole concentration in the collector

q electronic charge

R(vs,f) responsivity

R equivalent input noise resistance of the preamplifier

R source resistance at the input of the priamplifier

S surface recombination velocityy

SNEP specific noise equivalent power

T absolute temperature

T measured time constant of the emitter junction

e .

tb base transit time

VBE base-emitter junction voltage

VCB collector-base junction voltage

x depth measured from the top surface

xB depth of the edge of the collector-base space charge

region on the base side

x depth of the edge of the collector-base space charge

region on the collector side

x total thickness of the phototransistor

Y admittance of the emitter-base junction

e

Y collector-emitter transadmittance

ce

W width of the electrical base region

a light absorption coefficient

(X fraction of the current injected from emitter to base

that reaches the collector

P a.c. current gain

YN emitter injection efficiency

Af bandwidth of spectrum analyzer

TN d.c. quantum efficiency of the base region

1p d.c. quantum efficiency of the collector region

TPTD total photodiode d.c. quantum efficiency

nPTX total phototransistor d.c. quantum efficiency

Tn a.c. quantum efficiency of the base region

Tptd total photodiode a.c. quantum efficiency

X optical wavelength

-4

micron = 10 cm

un electron mobility in base region

v optical frequency of the signal radiation

Tn electron lifetime in the base region

cM radian electrical frequency

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 IN' PHOTOTKANSISTORS

BY

Fralcisco II. De La ::unedd

August, 197U

Chairman: Dr. Eugene R. Chenette

Major Department: Electrical Engineering

SThe investigation reported here presents both analytical and experi-

mental results on the noise performance of phototransistors. The noise

performance of phototransistors depends on fluctuations traceable to two

main sources: a) The random fluctuations in the rate at which photons

are received by the phototransistor; b) Mechanisms inherent in the device

such as fluctuations in the generation of free carriers, diffusion fluctua-

tions, recombination fluctuations, and 1/f mechanisms, to cite a few. On

the basis of corpuscular arguments, it is shown that the important fluctua-

tions may be represented by simple, partially correlated shot noise current

generators in parallel with the junctions for a wide range of frequencies.

The hybrid-pi transistor model and these noise generators have been used to

derive an expression for the noise current appearing at the output of a

phototransistor. Attention has been focused primarily on the low-frequency

expression since correction terms due to high-frequency effects are

negligible.

The output noise has been referred back to the input in order

to obtain the NEP and D To compute the NEP requires determining the

quantum efficiency of the collector-base photodiode. Previously published

analytical expressions seem to be inadequate for devices with good base-

region recombinative properties. Therefore, a numerical approach has been

implemented to compute the quantum efficiency of the phototransistor base

region. The computed quantum efficiencies agree well with published data.

The output noise spectra of commercially available silicon photo-

transistors and silicon units fabricated at the Microelectronics Laboratory

of the University of Florida have been measured. These measurements are

in good agreement with the predictions of the theoretical results. The

current gains and beta cut-off frequencies of these devices have been

computed by interpreting the measured spectra with the aid of the analytical

expressions. Independent a.c; and d.c. measurements of these parameters

are in satisfactory agreement with the noise results. It is thereby as-

certained that measurements of phototransistor noise can be used to

characterize them in terms of current gain and beta cut-off frequency.

xiii

CHAPTER I

INTRODUCTION

The phctcLransistor was first suggested in 1951 by Shockley, Sparks

and Teal as a variation on the "hook" transistor. It stayed in a dormant

state until recently '.hen considerations of cost, size, reliability and

the planar semiconductor technology have catapulted it into a prominent

position in the imaging sensing area. Its potential to nmke possible

low cost, long life, low power-consumption sensing arrays open up ne,' areas

of application where the traditional electron beam systems cannot compete.

Moreover, with the advent of GaAs diodes which radiate light in the infra-

red range, the silicon phototransistor has acquired added significance.

It is the natural complement of these light-emitting diodes on account

of the good responsivity of silicon to radiation in the infrared. Thus,

a new technology known as optoelectronics has been developing for the

past five years. Its mainstay sensor is the silicon phototransistor for

frequencies up to about 1 Mhz.

An emerging technology needs characterization of its components so

that their advantages and trade-offs with respect to what other technolo-

gies offer can be better determined. For the case in hand, it is obvious

that characterization of the silicon phototransistor is of prime importance.

Several workers in the field have already carried out part of the task.

Among recent valuable contributions, the following are relevant to the in-

vestigation undertaken in this paper: Gary and Linvill2 have derived analyti-

cal expressions for the quantum efficiency of a photodiode and developed

a model for optical phenomena in diodes and transistors; Joy and Linvill3

have analyzed phototransistor operation in the charge storage mode;

Schuldt and Kruse have treated the problem of image resolution of a single

phototransistor illuminated by nonuniform light; finally, a whole issue of

the IEEE Trans. on Electron Devices5 has been dedicated to the subject of

solid-state imaging. The subject of noise in phototransistors was touched

upon by Daughters. However, his analytical treatment was brief and no

attempt was made to fully explain the experimental data. Thus, a complete

study of noise in phototransistors involving both analytical and experi-

mental work remains to be done if the task of fully characterizing photo-

transistors is to be finished. The results will establish the fundamental

limit to its use as a radiation detector and amplifier. Herein lies the

principal source of motivation for the investigation to be undertaken. A

second source of motivation, by no mcans secondary in importance, lies with

the nature of noise itself. Noise measurements give second order moments

of the current and voltage fluctuations in a device; whereas most other

measurements give first order moments or averages. Hence, noise measure-

ments reveal in finer detail the physical mechanisms that govern the be-

havior of charge carriers inside the device. Therefore, it is to be ex-

pected that new insights may be gained into phototransistor operation

through noise measureuents.

An expression for the spectral density of the short-circuited output

current has been derived including high-frequency effects. This expression

has been verified with noise measurements and other independent measure-

ments. Both commercial phototransistors and units made at the Microelectronic5

Laboratory of the University of Florida have been tested. It is thereby

shown that noise measurements on phototransistors can be used as a tool to

characterize them, a result of importance particularly if the base lead is

not available.

To complete the task of noise characterization, the output noise has

*

been referred back to the input using the concepts of NEP and D To

accomplish this the quantum efficiency of the phototransistor must be

known. The results of Gary and Linvill have been improved upon by means

of a numerical solution of the pertinent equation. It is found that their

formula can be in considerable error for the tange of lifetime values

associated with good phototransistors.

It is hoped that this new method of phototransistor characterization

presented here for the first time will help to create new approaches to

the solution of the problems arising from the nonuniform distribution of

hFE among the phototransistors that constitute large sensing arrays.

CHAPTER II

ANALYSIS OF D.C. AND A.C. PHOTOTRANSISTOR CHARACTERISTICS

One of the difficulties inherent to noise studies is that the analy-

sis combines simultaneously both the d.c. and a.c. characteristics of the

device under scrutiny. This investigation of noise in phototransistors is

no exception to the above general observation. Thus, the first efforts

will be directed toward understanding the fundamentals of phototransistor

operation to obtain the d.c. and a.c. equations needed for the noise analysis.

D.C. Characteristics

A description of how the phototransistor functions will permit us to

establish which mechanisms are of first order importance for the type of

device geometry envisioned. As these mechanisms are explained, plausible

assumptions for the ensuing analysis will be stated.

As in the case of the transistor, transport of minority carriers and

their recombination in the base region play a central role in the operation

of the phototransistor. In addition, photogeneration of minority carriers

and their majority carrier counterparts is as important. Irradiation of

the top surface of the planar phototransistor shown in Figure 2.1 by a

uniform photon flux, F(v), causes generation of excess majority and minority

carriers throughout the whole device. Therefore, both junctions act as

photodiodes.- However, usually the collector-base junction area is one

order of magnitude larger than the emitter-base junction area. Moreover,

the photon flux per unit area penetrating the base region is much larger

than that going into the emitter because of light reflection and absorp-

tion by the emitter metal contact. Together these two factors relegate

the emitter junction to a secondary role as a photodiode. Consequently,

the photogeneration rate in the active base region, region I in Figure

2.1, will be neglected and only that corresponding to the surrounding region

II will be taken into consideration. This means that the phototransistor

can be sectioned into a one-dimensional collector-base photodiode in

parallel with the transistor section. This modeling is shown, schematically,

in Figure 2.2a. A better approximation would include a resistive element

between the photodiode and the base of the transistor to take into consid-

eration self-biasing effects at high-injection levels, as shown in Figure

2.2b. For the current density levels that will be treated in this investi-

gation this effect is negligible unless the base is very lightly doped.

What follows is based on the assumption. that the model of Figure 2.2: i

adequate.

Figure 2.5 shows the currents components for an n-p-n phototransistor

with its collector-base junction reverse biased. It will give us insight

to examine in detail the physical processes that bring about these currents

from the moment that the incoming radiation turns on until steady state

conditions are established. Excess pairs of carriers are continuously

generated on both sides of the collector-base photodiode. The emitter

and collector potential barriers box in the excess majority carriers in

their respective sides. On the other hand, the excess minority carriers

within one diffusion length of the junction edge glide over to the other

E C

N P

'r region II

reaon I w x

N xc

Figure 2.1 ross-sctional viw of planar npn potr

Figure 2.1 Cross-sectional view of planar n-p-n phototransistor

(a)

(b)

Figure 2.2 One-limensional models of phototransistors. a) For

low-injcctLon 'lvel. b) For high-Lnjeccion level

FE)( IPH

qv _E

a e kT

oeE /

I ekT-

E q VBE

IET e

,"/ IB=0

mV /

/

/

ID(N )e e

YN

N emitter

Ip \ \

\ \

P base

Legend

*-, electron

o hole

,^ photon

^v phonon

q VB E

aF ID ekT

JOC

N collector

Figure 2.5 Bulk current conpone;ts for an n-p-n phototransistor.

Carrier sources and sinks are also shown

side of the junction and give rise to the photogenerated current IpH.

From Figure 2.3, it is clear that this current is made up of electrons on

the collector side of the junction, IN, and of holes on the base side,

Ip. Obviously, IpH = IN = Ip. In addition, the leakage component, IBC'

due to thermal excitation is also flowing through the collector-base

junction. Even though each photoionization event creates a hole-electron

pair, the flow of current IpH initially sustains the build-up of a positive

hole charge in the base region. This charge neutralizes some of the fixted

impurity charge at the edge of the junction space charge region with a

subsequent lowering of the potential barriers. This has practically no

effect on the collector-base junction which is externally biased. On the

other hand, the lowering of the emitter barrier is sufficient to induce

injection of minority carriers across it. These injected carriers either

recombine with the excess majority carriers or neutralize them while they

reach the collector. Hence, an equilibrium is reached when the rate of

majority charge growth in the base is equal to the rate at which recombina-

tion takes place since the neutralizing minority carriers stop the photo-

generated majority carriers from further lowering the emitter-base poten-

tial barrier. In conclusion, the current (IBC + IpH) plays the role of a

base current source which biases the phototransistor and sustains the

losses of minority carriers in the base.

Gain Mechanism

The collector current is made up of the current (IBC + IPH) plus the

fraction Oa of the current injected by the emitter which reaches the

collector.

IC =,oIDexp(qVB/kT) + (IBC + IH) [2.1]

The lowering of the emitter barrier induces injection not only from emittc.r

to base but also from base to emitter. It is convenient to introduce the

familiar concept of emitter efficiency, yN, to relate these two currents.

In addition, leakage currents due to bulk and surface defect centers also

flow across the emitter junction. This component is usually represented

by mans of a diode with a non-ideal exponential dependence, i.e.,

qVBE/m'kT for m' / 1. For moderate levels of operation, the total emitter

current is

I

IE = exp(qV /kT) + IETexp(qVBE/m'kT) [2.2]

7N

The net base current must be zero. This constraint and Eqs. [2.1] and

[2.2] yield

ID (IBC + Ip1) IFTexp(aVBE/m'kT) 2.]

N )7exp(qVBE/kT) = (I YN o)

This expression clearly shows the deteriorating effect of the emitter-

leakage current on the device gain. Nevertheless, for normal operating

levels, a substantial gain is in effect to give the phototransistor a big

edge over the photodiode. Substituting Eq. [2.3] into Eq. [2.2) gives

an expression for the total output current, ICEO. The subscript here in-

dicates that IC = I = ICE with the base open

ICEO 1 [(IBC + IH) 7N oF IETexp(qVBE/m'kT) [2.4]

It has already been pointed out that (IBC + IpH) plays the role of a base

current source. Hence, by rearranging this last equation to yield the

ratio ICEO/(IBC + IH), one obtains (hFE + 1), the d.c. current gain

-1 7N 'F IETexp(qVBE/m'kT)

(hF + 1) = (1 N ) + I 2.5

C EO

Quantum Efficiency

Definition

The d.c. current gain (hFE + 1) has been expressed in terms of quanti-

ties which are not directly measurable, namely IPH and IBC. Therefore,

(hFE + 1) cannot be directly measured ( see-Chapter III for a method.)

However, the input photon flux which causes IpH is directly measurable.

Hence, if it is desired to express the d.c. gain in terms of measurable

quantities, the concept of quantum efficiency must be introduced. The

quantum efficiency of a phototransistor, ,TX, gives the number of output

electrons per incident photon

TTX CEO [2.6]

qF(v)Ab

In order to insert (hFE + 1) into this equation, it is necessary to elim-

inate the contribution of IBC to the output current since we are concerned

here only with output electrons associated with the photogeneration process.

Experimentally, this condition corresponds to keeping VCB = 0.0 volts.

Let (hE + 1) correspond to this condition, then

PTX = E H = (~E + l)TPTD [2.7]

qF(v)Ab

where IPTD is the quantum efficiency of the collector-base photodiode of

Fig. 2.2a. Eq. [2.7] shows how the photodiode quantum efficiency is in-

creased by the current gain of the transistor.

Analytical solution of the quantum efficiency

Gary and Linvill2 have analytically treated the problem of the quantum

efficiency of the collector-base photodiode. The authors introduce quantum

efficiencies N, and rp for hole-clectron pairs generated in the base and

in the collector, respectively. To obtain them, these authors have

solved the continuity equation for minority carriers,both in the base and

collector regions.

For the base region a Gaussian inmpurity distribution is assumed. For

reasons of mathematical simplification, the corresponding drift field is

set equal to the average electric field, E. With this approximation, the

continuity equation in the base region reduces to

2 dn

dn n

2 ( ) d

dx n

___p n C F(v) exp(-Cx)

2 D

L n

n

[2.8]

when this equation is solved subject to the boundary conditions

J = qnS at x = 0

n

n n = 0 at x = x

P B

[2.9]

[2.10]

the resulting base-region quantum efficiency is

TN = 2 2

N = 2Na 1/L2

P

x I(- N)[(s H N)exp(-a

**
(s M N)exp(-MxB) (s + M N)exp(-L' B)**

( + N) [(s + C 2N)exp(-H N)xB (s + M N)exp(-Oc**
**

(s H N)exp(-IxB) (s + M N)exp(NlXB)

0B B

(2N Q)exp(-cc**
**

[2.11]

where

s = S/D

n

N = qE/(2kT)

M = (N2 + 1/L2) 1/2

n

The collector contribution to the total quantum efficiency is easier

to compute since,for the type of device under consideration, the collec-

tor doping is uniform with no drift field. An equation similar to Eq.

[2.8] but without the field term is obtained for the minority holes of the

collector. This equation is then solved subject to the boundary conditions

p p = 0 at x = xC and x = xS to obtain collector quantum efficiency.

Oexp(-QxC) x-xC exp[(xx] [2.12

TIP 2 2 L L L

ip 2 l/L --_ coth ( ) + I c [2.12]

( 1/L) p ) (xs-xc)

sinh

L

p -

The total quantum efficiency of the collector-base photodiode is, of

course, given by the sum of Eqs. [2.11] and [2.12]

DPT = TN + P [2.1]3

Since Eqs. [2.11] and [2.12] are rather cumbersome, their authors have

had to use a computer to evaluate them for different combinations of para-

meters. From the 12 resulting plots, they have extracted some physical

interpretation. They also show a comparison between values of TPTD

obtained theoretically and experimentally. Invariably, the former are

larger than the latter. The discrepancy gets progressively worse as the

optical wavelength shortens.

Numerical solution of the quantum efficiency

It is apparent, from the complexity of the analytical solution and its

failure to match experimental values, that an improved solution for q

can be obtained by eliminating the assumption of a constant electric field.

The pertinent equation will then have to be solved numerically. This is

not really a great inconvenience since Eqs. [2.11] and [2.12] had to be

evaluated with a computer anyway. The advantages to be gained are numerous.

In addition to doing away with the assumption of constant electric field,

the numerical solution allows one to use any kind of desired impurity

profile whether it be analytical or experimental. Variations of drift

mobility with impurity concentration can be easily incorporated. This is

an important point since the value of the mobility that is of importance

is related to the doping near the junction edge. However, since the doping

is continuously changing, it is difficult to assign a particular value to

the mobility. A further refinement is to include variations of lifetime

with impurity concentration, although the small volume involved does not

warrant this effort.

The one-dimensional equation for the divergence of the minority carrier

current density in the base region is

1 dJ n n

S n -- = -r F(v)e:
q dx

The electric field is obtained from the constraint that J : 0 and that

P

for low injection levels charge neutrality is satisfied if the majority

carrier concentration is equal to the acceptor concentration C(x)

1 dC

E =(kT/q)( x) [2.15]

Substituting E into the well-known expression for J and, in turn, J into

n n

Eq. [2.4] yields the most general continuity for equation for small in-

jection conditions

d2n dD D dC dn

____ n n

D + -+ -- +

n 2 x C dx dx

1 dC n d 1 dC 1

Cnd+ n

C dx dx n dx C dx

n

CF(v)exp(-ox) [2.16]

This equation has been solved numerically by reducing it to two simultane-

dn

ous equations of first degree in the variables n and n subject to the

boundary conditions given by Eqs. [2.9] and [2.10]. Mathematical details

of the solution are given in Appendix A. The computer program given in

Appendix A can handle both Gaussian and complementary error function types

of impurity profiles. In the results presented below, the Gaussian option

has been used. D and dD /dx have been obtained from the curve fit tu

n n

experimental values of [n given by Caughey and Thomas.7

The total quantum efficiency of device 1 in Gary and Linvill's paper

has been computed using Eqs. [2.11] and [2.12] and compared against the

results obtained from the numerical solution for the base-region and Eq.

[2.12] for the collector. The comparison is shown in Figure 2.4 where the

-9

base lifetime is 1.7x10 sec. and the surface recombination velocity is

4x10 cm/sec. Both answers agree very well with each other, but they are

both well below experimental values. Measurements by Rosier give a

maximum S of x104 cm/sec. Hence, another comparison was made with S de-

creased to 2x10 cm/sec. The results are shown in Figure 2.5. It is

noticed that Eqs. [2.land [2.12] give much higher values of TID than

the numerical method for k <.7u. In fact, for the set of parameters used

.5

4

SPTD

.2

x T S

I "9 5

1.75/L 1.7x10 sec 4x10 cm/sec

a experimental

o numerical solution & Eq.(2.12)

a Eqs.(2.11) (2.12).

0 .0 1 1 1 1 ,

.4 .5 .6 .7 .8 .9 1.0

X (.L)

Figure 2.4 Computed and experimental collector-base photcdiode

quantum efficiency

.8

.7

.-C

.5

SPTD

.3

.2

x T

J n

--- 1. 75;

.1 1.75/J

-9 5

10 lO9soc 4x10 cm/sec

1.7xO1 soc 2xi0 cm/soc

o numerical solution & Eq. (2.12)

A Eqs. (2.11) E (2.12)

0.0I

.4 .5 .6 7

X(IL)

.8 .9 1.0

Figure 2.5 Computed collector-base photodiode quantum efficiency

As .

&, -,,

I \

I

S/ ,a \

I \

/ f^ "^ ~-< s

the numerical solution is practically insensitive to variations in S. On

the same figure, a comparison is shown for S at its original high value of

4x105 cr/sec and the lifetime increased to 10x109 secs. Again Eqs. [2.11]

and [2.12] yield much higher values chan the numerical solution although the

latter does show a noticeable increase above the values of Figure 2.4.

The increase is such that this last numerical answer agrees vary well with

the experimental data on Figure 2.4. Figure 2.6 shows plots for both S

and n changed to the new values: S = 2x10 cm/sec, T = 10x-9 sec. The

n n

answer from Eqs. [2.11] and [2.12] practically blows up at the shortest

wavelengths, whereas the numerical answer varies very little from the one

-9

with T = 10x10 sees. The conclusion is then that for X < .8u, Eq.

[2.11] is very sensitive to changes in both surface recombination velocity

and minority carrier lifetime while the numerical solution is only sensi-

tive to the latter. Only when both S and n are effective carrier killers,

both approaches agree. However, that is a rather undesirable set of param-

eters for good phototransistor performance.

These observations nay be qualitatively explained as follows: the

analytical solution assumes a constant electric field throughout the base

region which sweeps away extra excess carriers more effectively than the

variable electric field of the numerical solution. In particular, the

latter is practically insensitive to changes in S, a direct consequence of

the fact that the electric field vanishes at the surface for the Gaussian

profile used in these computations. For the near-infrared wavelengths,

both approaches give similar answers since at these longer wavelengths

most of the absorption takes place near the junction where the assumption

of a strong electric field holds better. The effect of a constant electric

field becomes negligible only when the excess minority carriers recombine

1.0

.9

.8

.7

.6-

"7PTO-5-

.4

.2S

1.75p. 10x10 sec 2xIO crn/sec

-1 experimental

onumorical solution a Eq. (2.12)

aEqs.(2.11) a (2.12)

0.0 1 1 1 1

.4 5 .6 7 .8 .9 1.0

X(/ )

Figure 2.6 Computed and experimental collector-base photodiode

quantum efficiency

quickly at the surface or/and in the bulk which corresponds to cases of

high S or/and lo Trn respectively. It is then that the analytical solu-

tions are trustworthy; otherwise the numerical approach is called for.

Using the values of S and rn of Figure 2.6, plots of IPTD were com-

puted numerically for devices 2 and 3 of Gary and Linvill's paper. These

plots along with their experimental counterparts are shown in Figure 2.7.

The agreement is very good over the whole spectrum.

A. C. Characteristics

Transistor Small Signal Equivalent Circuit

Small signal equivalent circuits have long been worked out for

transistors of different fabrication types under a diversity of operating

conditions. From this large supply, the model which makes the noise

analysis easiest should be selected. As it turns out, such model is the

hybrid-pi. Its well-known schematic diagram is shown in Figure 2.8a

along with a collector-base excitation source representing the photodiode.

Figure 2.8b shows an equivalent representation.

A. C. Quantum Efficiency

The collector-base photodiode has been characterized by means of its

d.c. quantum efficiency. The nurmrical method used for the d.c. case can

also be extended to find the a.c.- quantum efficiency, Tn, of the base-region.

Under time-varying conditions, Eq. [2.16] has a new term on its right-hand

side.

-O- -k \4

/I

/ '

I,

I,

. / ii

II

.2 I

i / x

l /J

2.7p.

.I ---. 4.0O

T S

n

IOxlO(sec 2x103 cm/sec

IOxl09soc 2xl03 cm/soc

o numerical solution a Eq. (2.12)

o experirenial .

0.0 ,

4 .5 .6 .7 .8 .9 I.

X (/[.)

Figure 2.7 Computed and experimental collector-base photodiode

quantum efficiency

7P TD

Iph

;-I(--

e

(a)

b

ph r, C b ph

e

(b)

Figure 2.8 Transistor equivalent circuit. a) Hybrid-pi model

with collector-base excitation source. b) Equiva-

lent configuration

2 dD D

D n(x,t) n n dC n (x,t)

n 2 dx C dx x

1 dC dDn d 1 dC 1

+ dx dx n dx" C( dx) n(xt) =

n

dn (x, td

nxt) Caj (vs,f)exp(-Qx) [2.17]

we separate variables by letting

n(x,t) = n(x)exp(jut)

After inserting this type of solution into Eq. [2.17] and regrouping, the

equation to be solved becomes

d n dD D dC dn

D + n n

n 2 dx C dx dx

dx _

1 dC dn d 1 dC (1 + jurn

+D -T n

C dx dx + n dx C dx

j. (vs,f)exp(-Qx) [2.18]

By inspection of this equation and Eq. [2.16], it is apparent that for the

a.c. case, n/(l + jo n) replaces n. If the analytical results of Gary

and Linvill are used, it then suffices to replace Tn by T /(1 + jaWn) in

Eqs. [2.11] and [2.12]. Since n is normally in the nanosecond range,

frequency must be well into the megahertz range for Tn to be significantly

different from its d.c. value.

The numerical solution of Eq. [2.18] involves a larger effort which is

sketched next but not carried out. Since the coefficient of n(x) in Eq.

[2.18] is a complex number, the solution for n(x) will also be complex.

By replacing n(x) with a complex number, one can obtain from the original

equation two new equations corresponding to the real and imaginary parts

of n(x). The same algorithm used to find qN from Eq. [2.16] is then

applied to these new equations to find the real and imaginary parts of rn*

Responsivity

Eq. [2.6] relates the output d.c. current, ICEO, to the input photon

flux. A similar relationship can be written for a.c. signals using the

concept of responsivity R(v,f) of the phototransistor. Its definition is

is (s,f) = R(Vsf)hvj s( s f)Ab [2.19]

where hvs (vsf)Ab is the r.m.s. value of the radiant pouer incident on

the phototransistor area Ab. The units of R(vs,f) are amps/watt. The

incident a.c. photon flu:x, js (v,f), excites a small signal current in the

collector base photodiode as indicated in Figure 2.8a. The equivalent

representation of Figure 2.8b clearly shows that the excitation is ib = ph.

The short-circuited output current is

is = (6 + l)ib = (P + )qlptds(Vs ,f)Ab (2.20]

By comparison of Eqs. [2.19] and [2.20], an expression for the responsivity

is obtained

R(vs f) ( 1ptd [2.21]

hv

s

CHAPTER III

NOISE CHARACTERIZATION

Since the phototransistor operates as a combination of energy trans-

ducer and amplifier rather than just as an amplifier, the figure of merit

conventionally used to characterize the latter's noise performance has to

be discarded in favor of a more fitting quantity. Thus, the concept of

noise equivalent power, or NEP, replaces the familiar equivalent input

noise resistance. The NEP is defined as the r.m.s. value of the sinusoidally

modulated radiant power falling on the detector which will produce an r.m.s.

output signal current equal to the r.m.s. noise current from the detector.

This definition can be written in a way which makes clear the relationship.

NEP hv s (vs) b (5.1f

.2 .2

Vi 1I

0o s

This equality implies that if one knows the right-hand relationship be-

tween the output signal current and the signal power excitation, then one

can find a "lumped" noise power which causes the output noise current.

This definition is more commonly written as

NEP = hvs s(v,f)Ab [5.2]

.2

2S

In the responsivity given by Eq. [2.21], we have a relationship between

the output signal current and the signal power. Thus, it is only necessary

to find an expression for the output noise current in order to put Eq.

[3.2] into an explicit form. Deriving this expression constitutes the core

of this chapter.

The double nature of a phototransistor as a power sensitive device or

transducer and as an amplifier motivated the discussion of the NEP. Then,

it is natural that we take this salient feature as a starting point for

the following discussion on sources of noise in phototransistors. Power

sensitive devices and amplifiers are the two main categories into which

radiation sensitive devices may be classified. Their noise mechanisms

are quite different too. In power detectors the noise may be considered

to arise from the random. arrival of individual photons. In amplifiers

the input signal may be considered noiseless but the amplifying mechanism

adds noise power to the signal. Therefore, it is our objective to investi-

gate these two different sources of noise when they appear combined in a

phototransistor

Collector-Base rlotoulode N;oiae Curr tiL

Since the noise in a power detector depends on the randomness with

which the incoming photons arrive, it is evident that the induced noise

depends on the type of light source. In the experiments to be described

in Chapter IV an incandescent tungsten lamp bulb has been used. In pra-

tice, daylight is the commonest source of radiation that phototransistors

are exposed to. Both of these sources are good examples of blackbody

radiators. A lucid treatment of this problem is given by van Vliet.

The basic idea for the case of thermal or blackbody noise is that in

1 cmr the number of photons in modes within a unit frequency interval Lv

fluctuates about an average value given by

S= 8 v2 Av [5.3]

c [exp(hv/kT) 1]

with a variance

2 1

varN= ( = N + exp(hv/k) = N( + B) [35.4

where B is the Boson factor. Notice that var N > N in contrast with a

poissonian process where var N = N. Therefore, the photon flux F(v) due

to the average number given by Eq. [5.3] carries a noise power larger

than shot noise. The extra noise NB is called wave-interaction noise for

the following reason: in any physical system there are losses and the

normal modes will be resonances of finite bandwidth. The random inter-

ference between the different spectral components within this bandwidth

cause the extra noise, NB. The Boson factor becomes significant only for

XT > 1.5 cm-Ko which involves extremely high temperatures or/and rather

long wavelengths. Since neither of these circumstances is contemplated

in this investigation, B will be neglected. Since 1PTD < i, not all photons

will produce output electrons, and consequently partition noise arises at

the output current due to the random misses. With the aid of the variance

theorem, one gets the variance of the excited carriers. Here, n stands

for both electrons and holes in the base and collector regions respectively,

var n = (qPTD 2 N + N(PTD PTD)

or

var n = PTDN [5.5]

Hence, the variance in the number of excited carriers gives a shot noise

of a strength corresponding to the photocurrent IPH = qPTD F(v). In con-

clusion, the fluctuations in IpH can be represented by a shot noise

generator located across the collector-base junction

.2

iph = 2qlpHtE [ 3.6

As previously discussed, in the section of Chapter II de-oted to d.c.

characteristics and shown in Figure 2.5, the only difference between the

currents IPH and IBC is that the former is due to photon excitation

while the latter is due to phonon excitation. The statistics of these

two types of particles are the same. Consequently, the noise of the dark

current IBC is also pure shot noise

bc = 2qlC

This shot noise generator must then be connected in parallel with the

generator of Eq. [5.6] across the collector-base photodiode.

Output Noise Current at Low Frequencies

In this section we fully develop the phototransistor noise model

taking into account the photodiode shot noise generators given by Eqs.

[5.6] and [5.7) and additional generators associated with the amplification

mechanisms of the phototransistor. The analysis culminates by formulating

from these generators and the gain of the phototransistor an expression

for the output noise current.

As in the case of the conventional transistor, it will be assumed

that the passage of carriers across the junction potential barriers gives

full shot noise. In view of Eq. [2.1] for the collector current, and the

results obtained for the noise of IpH and IBC, we can write for the noise

of the collector current

i = 2q[IDexp(qVBE/kT) + (IBC + Ip)]Af [3.8]

Similarly, Eq. (2.2] can be used to write the noise of the emitter current

2 ID

i = 2q[ exp(qVBE/kT) + IETexp(qVBE/m'kT)]Af [5.9]

Reddi has analytically and experimentally shown that surface recombina-

tion through a single level generation-recombination center explains the

behavior of the leakage component of the emitter current. Lauritzen1

has analytically proven that this type of mechanism has a spectrum with a

strength varying from full shot noise at very low voltage bias to three-

fourths of full shot noise for VBE > .3 volt. Then, it does not introduce

great error to assume full shot noise for the leakage current as we have

done in Eq. [5.9].

At low frequencies one can neglect the base transit time, tb, and

consider that the current pulses of those injected electrons that do not

recombine appear simultaneously at the emitter and collector junctions.

The correlation between the emitter and collector noise currents is then

given by

i = 2q [ O IDexp(qVBE/kT) ] Af [5.10]

We can thus represent the noise by two noise current generators, i and

ic, across the emitter and collector junctions respectively. This

corresponds to the physical -T transistor model.

The fact that only the fraction OFIDexp(qVBE/kT) of the total in-

stantaneous emitter and collector currents is composed of simultaneous

pulses leads to the conclusion that there exists an instantaneous, cr a.c.,

base current inasmuch as i / i This instantaneous base current is a

c e

direct manifestation of the fact that (IBC + IpH) behaves like a d.c.

base current source. Thus, we can equally represent the noise by a

current generator ib = i -i in parallel with the emitter-base admittance

b c e

and a current generator i in parallel with the collector-emitter terminals.

This representation is evidently patterned after the hybrid-pi equivalent

circuit of Figure 2.8b; i2 is given, as before, by Eq. [5.8]. The mean-

squiirc value of ib is

2 *

S= (i i )(i i) ii + i -i i i i [3.11

b c e c e c e e c e e c

Substituting Eqs. [5.8], [5.9], and [5.10] into this last equation yields

ib 2q ( 1 "4 ) IDexp(qVBE/kT) + IEexp(qVBE/m'kT) + (BC +

Ip1) laf [5.121

_I

Using Eq. [2.5] it is straightforward to show that the first two terms in

this last equation equal (IBc + IpH). Physically, this is apparent since

the first two terms are the fraction of the emitter current that compen-

sates (IC + IpH) Thus, we have the result that the base current generator

has twice the shot noise corresponding to (IBC + IH); that is

i = [2(IBC + I)] [+.1]

Physically, this comss about because ib is r.ade up of the fluctuations due

to (IBC + Ipl1) and to that portion of the emitter current that corpensapes

(IBC + IPH); na-mely, the first two terms in Eq. [5.12]. Thus, these two

currents flow in opposite directions and cancel through recombination in

the base region, whereas their shot noises are independent and consequently

add quadratically.

It is illuminating, as well as useful, to compare this situation with

the case of a conventional transistor; in a transistor, the majority carriers

sustaining ib are supplied through an ohmic contact where no energy barrier

is involved. Therefore, the shot noise associated with the collector-base

barrier disappears and the term IpH vanishes from Eqs. [5.121 and [5.15]

causing a reduction of 5 db in the noise power of i if IBC is negligible.

b BC

Finally, the correlation between ib and ic is

.2

ibi = (i i )i = i i i [5.14]

be c e c c ec

Using Eqs. [3.8] and 5[.10] in [3.14] gives

ibc = 2q(IBC + IH)f [3.15

This result should be expected since (IBC + IpH) is common to both the

collec-tor a id as= c urnts.

The hybrid-pi equivalent circuit of Figure 2.8a is next used to

accommodate the just derived shot noise generators. The resulting noise

equivalent circuit is shown in Figure 3.1. The mean-square value of the

short circuited output current is obtained from this equivalent circuit.

.2 2 2 2 2 *

o =(b c = I b c + 2Re(bc)

where B = hfe/(1 + jf/f ). With the aid of Eqs. [5.8], [5.15] and [3.15],

fe B

Eq. [3.16] becor-.s

2 / (I + IpL \/2n-e + 2hfe

i = 2qlCE + iC + fe f)2 Af [

0 ICE + u i

Recalling that (hFE + 1) = ICEO/( C + PH) and using the fact that hfe is

C1

It b r TC,

1-1- C

T )9b C'C

E

2= 2q[aD Iexp(qVE/kT) (I IPAf

C F CP

i q 2 IC -I ) Af

b BC PHI

ib=

i =2q(

cb

I +I )Af

BC PH

Figure 3.1 Phototransistor noise model. ShoL noise

current generators i and i are imposed

on a hybrid-pi small signal equivalent

circuit

2h

usually sufficiently large so that h2 rhV[ 1

> > h Eq. simplifies to

.2 2he/hFE -

i = 2qlCEO 1 + f/ 2 Af [5.18]

o 1 + (f/f)2 )

B

This is the sought-after expression for the output noise current. This

approach is not fully correct for frequencies near the cut-off frequency

fT' where I8 = 1, but the device is not very useful there either. High-

frequency effects are dealt with in the next section.

The low-frequency plateau of Eq. [3.18] gives the shot noise of the

base current amplified by the gain of the phototransistor. As the gain

decreases with frequency, the output noise approaches a high-frequency

plateau equal to the shot noise of the collector current.

SAt a given ICEO, the unknown parameters of Eq. [3.18] can be deter-

mined from a measurement of the output current spectrum covering the range

from the low-frequency plateau up to some frequency above the 3 db fall-off

point. For the purpose of interpreting the measured spectrum, it is con-

venient to define an I by imagining that Eq. [5.18] gives the shot noise

of a frequency-dependent equivalent current I ; namely,

2hfe/hF

eq =CEO 1+ f 2 [5.191

I + (f/f) 6

The value of I corresponding to the low-frequency plateau gives the

parameter he/hFE. With this parameter known, it is then a straightfor-

ward matter to find f from the measured Ieq.

In summary, Eq. [3.18] not only gives the magnitude of the output

noise of a phototransistor, but it can also be used to characterize it by

2

means of the parameters h /hFE and f This can be a very useful tool

particularly when the base lead of the phototransistor is not available.

This topic will be fully treated in Chapter IV.

.Output Noise Current Including High-Frequency Effects

At high frequencies, when transit time across the base becomes important,

the real part, g,e of the complex emitter admittance, Ye, departs from

its low-frequency value, geo = IE(q/kT). This real part, ge, contributes

full thermal noise. Then Eq. [(.9] can be amended by simply adding the

correction terrc 4kT(ge g_) to it

2 = 2q ID

e exp(qV /hT) + Iexp(qV /m'kT) Af + 4-.T(ge -g)e f [).20

E BE 1 eeo

At high frequencies, Lauritzen's1 analysis predicts that the shot noise

of the generation-recombination leakage current is reduced by a factor of

2/5. However, since the defect-center relaxation time, for forward bias

conditions, becomes very small, this reduction would be only observed for

frequencies high in the megahertz range. Obviously, the device is not

very useful here, and this effect can be safely neglected. No other high-

frequency effects are attributed to this diode since no diffusion current

is associated with it.

Delay between emitter and collector current pulses due to the transit

time, tb, of minority electrons through the base region becomes important

-I 12

for frequencies comparable with (2t b) Following van der Ziel this

effect is introduced into the low-frequency equation for the correlation

between emitter and collector noise currents with the aid of the complex

collector-emitter transadmittance, Yce, which replaces the low-frequency

transadmittance in the term CF.I exp(qV /kT) = (kT/q)gc of Eq. [5.10]

i i = 2kTY Af [3.21]

c e ce

A glance at Eq. (5.11] indicates that i2 also changes on account of Eqs.

[5.20] and (5.21]. To simplify the algebra and obtain a better

physical insight, it is convenient to rewrite Eq. [5.20] as

.2

i = 2q IDexp(qVBE/kT) + (IBC + I Af kT(g )Af 5.22]

e C PH e eo

Substituting Eqs. [5.8],3[.21], and [3.22] into Eq. [3.11] yields

1 = 2q I20Iexp(qV./kT) 2 kTRe(Y_) Af + 4q(I,, + I,,) Af

L q

+ hkT(g geo)Af [5.25]

Recalling that YLIDexp(qVBE/kT) = (kT/q)gceo puts the bracketed term in a

clearer context

S= ceo Re(Yce) + q(IBC + H) + kT(ge eo) Af [5.24]

The interpretation of the first term is straightforward. At high fre-

quencies, the random delays that the injected minority electrons undergo

while diffusing from emitter to collector are taken into consideration.

These random delays add a new type of spontaneous fluctuation to the base

2

current and thereby increase ib. The nature of this noise is thermal

since diffusion is a thermal process. Hence this noise contribution is

given by Nyquist theorem, 4kTgAf, where g is the net conductance that these

electrons see, i.e., gceo Re(Yce). The last term of Eq. [5.24] has been

explained by van der Ziel as arising from injected minority electrons that

diffuse back into the emitter from the base. Hence, they add noise to the

base current since they look as if they were a thermo-leakage component

from base to emitter.

Lastly, the correlation between base and collector currents is ob-

tained after using Eqs. [3.8] and [5.21] into Eq. [5.14].

ibic = 2kT(gceo ce) + q(IBC + Ip) Af 1[.25]

By substituting Eqs. (3.8], [3.2h], and [5.25] into Eq. [5.16], the desired

noise of the output current at high frequencies is obtained

-- 2h /h

i = c2ql 1 + 2e f 2 f 4kT(g g )IA 2 Af

o CEO 1 + (f/f ) e

[5.26]

+ T ceo A R (gceo Yce) Af

The type of expressions used for the different admittances appearing in

this expression vary according to phototransistor type and geometry. In

Chapter IV, a special case will be studied to determine the relative impor-

tance of the high-frequency terms.

NEP. SNEP. and D-

We have expressions available nou for the noise of the short-circuited

output current of a phototransistor. These equations can be very useful to

the designer of the circuitry following the phototransistor without any

further modifications. However, the designer concerned with the optical

system anteceding the phototransistor needs this noise referred back to the

input; that is, he needs the NEP a.s defined by Eq. [3.2]. Now that we have

expressions for the output noise current, we can write the NEP explicitly in

terms of device parameters by using either Eqs. [5.18] or [3.26] for the noise,

and Eq. [2.20] for the signal current. For simplicity's sake, the low-fre-

2

quency expression for i is used

0

-- /2

2h2 /hE

1/2 1+ f

(2qICE) 1 + (fE )2 1/2

NEP = hv -- (Af) [5.27]

s q 1ptd h2 -

1 + fe

I + (f/f B

For frequencies up to hfe f, the inequality he > 1 + (f/f ) holds and

-e 6 efe

greatly simplifies Eq. [ 3.29]

NEP p2 hs CEO 1/2 1/2 [5.28]

lptd hFE

Using Eqs. [2.61 and [2.7] for ICE0 and recalling that ptd rPTD

for frequencies of practical interest, this last equation is further reduced

to

1/2

NEP 2h (FL ) (Af)1/2 [5.29]

s 'PTD

Thus, we have reached to the very important conclusion that the NEP of a

phototransistor is practically constant for frequencies up to f c h fe

As-Eq. [5.25] indicates, the NEP is directly proportion-l to

(AAf)1/2. By letting Ab = 1 cm2 and Lf = 1 Hz, we get the specific noise

equivalent power or SNEP. It is common usage to define the detectivity as

the reciprocal of the SNEP; that is

1

D = 1 13 .

S- SNEP (vs,f) [

The virtue of the D is that it has the psychological advantage of increas-

ing as the device improves noise-wise.

CHAPTER IV

EXPERIMENTAL RESULTS

The purpose of the conducted experiments, whose results are reported

in this chapter, was to verify the theoretical results of Chapter III and

to exprlre the usefulness of noise measurements as a characterization tool.

The shape of a measured spectrum and its asymptotic values give

sufficient information to make a fruitful comparison between theory and ex-

periment. However, the comparison is not entirely foolproof since there

may be spontaneous sources of noise not taken into consideration which dis-

tort the spectrum. For a conclusive check, we need to directly compare

2

hf /lE and f obtained from noise measurements with results obtained from

a.c. and d.c. measurements. This has been done by carrying out transit-

time5 and hF measurements on three-lead phototransistors.

To show that no inconsistency is introduced by comparing r.easured

parameters obtained under different sources of excitation, measurements

were also done using a GaAs light-emitting diode. The measured frequency

response of phototransistors to an amplitude-modulated light signal from

a GaAs diode has been used to determine f B

Noise Measurements

Introduction

It is well known that noise measurements require extreme care and

caution. -The problems encountered are manifold: noise from the passive

and active components of the preamplifier may mask the noise being measured,

strong 60-cycle pickup clips the noise waveform with resulting crossmodula-

tion products, pickup of extraneous electrical or acoustical signals may

adulterate the noise measurement, noisy components such as sockets, batteries,

or potentiometers may go undetected since they do not alter the expected per-

formance of the preamplifier and yet they may be playing havoc with the

noise measurement. These problems and others have been encountered during

the course of these measurements on phototransistor noise. Safeguards

against these problems are available. First, it is necessary to continuously

monitor the noise with an oscilloscope so that abnormal behavior does not

go undetected. A screen room is a standard technique used to shield noise

measurements from outside electrical signals. Unfortunately, at the time

of 'this investigation, one was not available. Improvisation, and the fact

that these measurements only covered the frequency spectrum up to 10 ITlz,

made it possible to work without a screen room.

A good number of the problems listed above are directly related to the

preamplifier. It is then a good investment to build a low-noise preampli-

fier whose characteristics and limitations are well known by its user.

After building several tube and transistor units, a cascode-cathode-

follower combination, using RCA 7586 Nuvistors, was decided upon. Its

advantages are several: high input impedance, large dynamic linear range,

compactness, relative imperviousness to large temperature changes, very

flat gain from 10 Hz up to 2 1Mz, and a low input noise resistance, R .

A plot of R vs. source resistance, R is shown in Figure 1.1. The low

noise resistance of 220 0 and measured gain of 52 db make this preamplifier

adequate for the needs envisioned.

-"o

0

u

0

So,

\ 3

TO 0..

a)

u

) o

I

Uo L-

3 r

L\i m

Q) L

a -4

1o -4

UUN

-4

Method of Measurement

As briefly discussed in connection with Eq. [).19], the objective of

these measurements ,jas to obtain plots of I vs. frequency. This was

eq

accomplished by measuring i

0

2

i = 2ql Af [(.1]

o eq

Since it was desired to measure a noise current, the phototransistor

was biased with a current source. Figure 4.2 shows the mcasureient set-up.

The voltage supply, VCC, biases the phototransistor through d 100 K P re-

sistor. With the device biased in this manner a comparison method was

then used to measure I eq. At a given frequency, two measurements were

needed in order to obtain Ieq. Let M (f) be the output meter reading

obtained after the amplified input noise was filtered through the band-

width ",f

1M (f) = (i2 + bg.2

where IGI is the power gain of the whole system and it is unknow.n,i2 is

0

2

given by Eq. [!.1], and i is the background noise. The second measure-

bg

ment was used to calibrate the system since [cG was unknown. In addition

to the noise terms in Eq. [h.2] another term corresponding to the calibrat-

ing signal was added to the input of the system preamplifier. At this

point it is convenient to digress from our topic to describe the types of

calibrating signals used.

For frequencies up to 1 MHz, a signal-generator-variable-attenuator

combination was used. The signal ",as delivered either by the H-P 502'A or

the 310A wave analyzer model operating in its BFO mode. This is a handy

feature of these instruments since their output signal is locked to the

center frequency of the bandwidth of their input tuned voltmeter. The

lo

Bcr

rr

w

N

z-J

0 0

N

0-

CC

LL l

in

ro)

LL

_=L:

Z

b--4

502A model covers the frequency range from 10 Hz to 50 KHz while the 710A

goes from 1 KHz to 1.5 MHz. The output of the wave analyzer was terminated

using the resistance R2 in series with the 50f1 characteristic impedance of

the variable attenuator which was in turn terminated by a 50o feedthrough

connector. A 100 KM resistor was inserted between the attenuator and the

preamplifier input so that the calibrating signal behaved as a current

2

source, i at the input of the preamplifier.

cal

For frequencies typically lying above 1 MHz a noise diode was used.

Since its output impedance was high, it was connected directly in parallel

with the phototransistor by tuning the load inductor, L, to resonate with

the stray capacitance of the jig.

The calibrating measurement, M2(f ), was normally done at a po.er

level several db above M (f )

1 o

2 2 .2 i lGi = 10n41 (f

M2(fo)= (i + bg + iCal = l E'(

2 o o bg cal

For the case of sinusoidal calibration, n = 3, for the noise-diode

2

case n = 3.53 Since the preamplifier used was low noise, i was negligible.

bg

Hence

.2

-2 cal 4.]

0 10 1

or in terms of I

eq

eq 2q Af (10n 1)

2

If a noise diode is used for calibration ica is given by

cal

cal = 2qINDw

and there is no need to know the effective noise bandwidth of the spectrum

analyzer. Since the majority of the noise measurements made during this

investigation used a sinusoidal calibrating signal, it was necessary to

determine the effective noise bandwidths of the two wave analyzers listed

earlier. This was done by replacing the phototransistor in Figure 4.2

with a noise diode and using a sinusoidal signal for calibration. The

measuring frequency was above the 1/f region of the diode spectrum, hence

Af was readily determined from Eq. [4.5] since Ieq IN. This procedure

gave a bandwidth of 5.25 Hz for the H-P 30\2 unit used in this laboratory,

while the manufacturer's figure is 6 lIz. For the H-P 310A, the bandwidth

was found to be 2.54 KHz vs. 35 Hz as given by the manufacturer.

Close to the end of this investigation, the General Radio type 1921

Real-Time Analyzer became available. The system performs the same func-

tions as the H-P wave analyzers; that is, filtering and measuring the r.m.s.

value of the filtered waveforms. However, the G. R. unit does this in real-

time over 45 channels covering the frequency spectrum from 5.5 Hz to 80 KHz.

Each input channel consists of an attenuator and a 1/5-octave filter. The

bandwidth of each channel was measured using the procedure described earlier

and found to be (.25' + .05)f Therefore, the input attenuators must be

adjusted in order to have the gain-bandwidth product of the system constant.

Having done this, one can then obtain an absolute measurement of the noise

over the available spectral range after calibrating just one of the 45

channels provided that the gain of the preamplifier is flat. Unfortunately,

due to the low output impedance of the phototransistor, the gain of the

phototransistor-preamplifier combination drops with increasing frequency

above 3 KHz.. Therefore, for the phototransistor noise measurements, the

measured spectrum had to be corrected to take into account the changes in

gain. Still, measurements with the G.R. system were twice as fast as with

the wave analyzers.

C

75

0

*-0.

w II N

> cr -L

0 o

O

..

t--

II

-C

II

I I

@-3

m +

4-

II

I I

A.C. Measurements

Transit-Time Measurement

The purpose of carrying out small signal measurements on these devices

was to check the values of he /hFE and f obtained from noise measurements.

Therefore, it was convenient to use an a.c. type of measurement which

simultaneously gave both quantities. Such measurement :.'a done using the

transit-time bridge.13 Figure 4.5 shows the bridge diagram. At the bal-

ance condition, a voltage null exists between collector and base and the

measured time constant is not affected by the collector-base capacitance,

CT. At a given emitter current, C and RC are adjusted for a null condi-

1

tion. When this is achieved, the following relationship holds1

T= % tb 4 (kT/ql )C (T .6]

e RB + RC b EET

fe RC

where CET is the emitter-transition capacitance. According to Eq.

[4.5], a plot of T vs. 1/IE has a slope of (kT/q)CET. This provides a

check on the measurement since CET can be independently measured. To find

14

f the expression derived by Lindmayer and Urigley was used

f

S= 1 [4.8]

6 kT

Sb + qE (CET + CCT) he

Hence, in addition to the quantities given by Eqs. [4.6] and [4.7], it

was also necessary to measure the collector-transition capacitance, CCT.

Frequency Response

The f of the tested phototransistors was also determined from the

3 db cut-off frequency of the device response to an amplitude-modulated.

light signal from a GaAs diode. The relevance of this measurement is that

it gives f under exactly the same physical conditions prevailing during

the noise measurement. Figure 4.4 describes the arrangement.

D.C. Measurements

The d.c. common emitter current gain, hFE, was measured using the

d.c. version of the transit-time bridge, hence VCB = 0 volts. However,

instead of finding hE from the ratio RB/RC, two R-P 425A d.c. ammeters

replaced RC and R and hence hE = I/IBg The internal resistance of

this ammeter is such that the maximum voltage across it is never larger

than 1 mV for all ranges. Then, for this two-meter arrangement, VCB is

still very close to null. Figure 4.5 shows the circuit diagram.

Noise Spectra

Noise measurements have been carried out on several commercially

available silicon phototransistors and on silicon units fabricated at the

Microelectronics Laboratory of the University of Florida. For each device

a family of curves was obtained for different values of ICEO and VCB = 0

CEO CB

volts, except for the T-I phototransistor type LS-400 whose VCB was set

at 15 volts. An incandescent light bulb was used for light source at all

times.

This is a two-lead device whose spectra were measured using the H-P

Texas-Instruments Phototransistor Type LS-400

1800P F

I

I

-_ I

- I1

ground

isolation

to

-b wave

Q analyzer

a

Figure 4.h Set-up used to nimasure the phototransistor f

B

using an amrplitude-Mrodulated-lighL signal

from a GaAs diode

2.2 Kn < c

<7

IE V O.O

4E AI H-P H-P

F 425A 425A

Figure 4.5 Bridge used to measure hFE

wave analyzers at low frequencies. For frequencies above 1 MHz, a noise

diode was used for calibrating source, and a Collins radio receiver for

spectrum analyzer. The measured spectra are shown in Figure 4.6 for

different currents. The low-frequency plateau predicted by Eq. [3.19] is

observed although it is masked by 1/f noise at the lowest frequencies.

The high-frequency plateau corresponding to full shot noise of ICEO is also

observed. The low-frequency plateaus are centered in the neighborhood of

100 KHIz. With the aid of the value of I at this frequency and Eq. [3.19)

eq

the parameter h/h has been extracted and plotted in Figure 4.7. These

values of he /hFE have been used in Eq. [3.19] to compute Ieq at f = f.

fe E eq 1

From such values of I and the spectra plots, the corresponding values of

eq

f have been determined. A plot of f vs. ICEO is shown in Figure 4.7.

The-plot shows f increasing with current. This is to be expected since

as the current increases, the emitter-base time constant, Te, in Eq. [4.6],

decreases toward the base-transit time, tb. These parameters have been used

to plot Eq. [5.19] over the whole frequency range in Figure 4.6. The agree-

ment is excellent.

Figure 4.8 shows spectra for ICEO = 20pA at different temperatures.

It is noticed that I converges toward 20 [A at 1 MHz. This corroborates

eq

that both the photogenerated IH and thermogenerated IB have the same spec-

tral density. If IBC is mainly due to recombination-generation centers this is

not true, the noise of IBC would be as low as 2/5 of full shot noise. How-

ever, since the contribution of IBC to the noise of ICEO is only a very

small fraction, the difference is practically undetectable.

Motorola Phototransistor Type MRD-510

This is a three-lead device and hence all the types of measurements

described before were carried on it. Its spectra, which were measured

"0

1-(

o

C)

o

S/CC

07

0 O O ON

c 'a,

U -

00 0 u

0 00 0 0 0 _

3I LO

0 .O

1 0

I -- I- II_ __ '

0o -o o o lo

(Vw) b I

E 0*

6~'I OQ cD

o ~J U.)

o o (ZHA ) o

0 0 0 0

*"4

w C

.0 0

-_ cO -

O

0 '

W U

ore

0 U

C)

o

\ \-0

I I ',

0 0 0 0

\) \

0 / C'

o o

re, c0

0

0

0

0

0

O

C-

O c

0

C

LL.

O

0

(V) bI -

(VUJ) 'I

52

with the 302A and 310A wave analyzers, are shown in Figures 4.9a and 4.9b.

Their shape is the same as that of the LS-400. However, it is a better

device since its low-frequency plateau is less masked by 1/f noise. By

2

the same procedure described for the LS-400 unit, the parameters h e/hFE

and f have been determined and plotted in Figure 4.10 and Figure 4.12. In

Figure 4.10, hfe from a.c. measurements (transit-time bridge) and hF are

also shown. The ratio m(IE) = hfe/hFE, which is a slow function of IE, is

useful in the discussion of these data. Plots of hFE, hfe = mhFE, and

2 2

he hFE = mhFE are available. m(IE) can then be computed by three

different methods. In Figure 4.11, m(IE) has been plotted by comparison

of noise and a.c. measurements, and via hfe/hFE where the comparison is

between a.c. and d.c. measurements. The good agreement between these two

independent approaches indicates that all these measurements are accurate

and verifies the low-frequency plateau of Eq. [1.19]. Figure 4.12 compares

values of f obtained from noise measurements, a.c. measurements, and the

GaAs diode experiment. The data agree within the limit of experimental error.

The good agreement obtained between noise and other independent mea-

surements and the convergence of I to ICEO at the higher frequencies

eq CEO

verifies the validity of Eq. [3.19] except where 1/f noise predominates.

Also, these measurements show the usefulness of noise measurements as a

tool to characterize two-lead phototransistors in terms of mhfe and .

Fairchild Phototransistor Type FPT-100

Using the G.R. system and the 310A wave analyzer, extensive noise

measurements were performed on four FPT-100 units. All measured spectra

bear resemblance to those shown in Figure 4.13 in that the low-frequency

plateaus are not too distinctly visible. This may be hinting that the

output noise spectra of these devices are not altogether controlled by

L:

//

- 0

o

7 0

rr)

O

0

-- 4

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S_.

0

SO O

C1

0

( u 0

'- 0

o 0

LL Li

S0o

o-Vi 0

0 .

iJ r 0

(vw) ")

(VUJ (Uo

0

0

0

c >i 0

o c

0 -

CA

>1

$,4

4-

0c o

E E 0 0 0 g ,

- 0r 10O C o

S(V ) I

0

afa

C.

0OaO

0 I I I I

0 00

4-

)E 0

0c

EE > 4

V) 0 o

0 0 C.

SE E

\0 0

w

o0 40 o >

I a V

E\i -

0

\ \ \ E- -u0

Cl c-

\\C:

Aj z

LO 0 LO 0 _O

r) 3 \\ D 0

-C LL

E E U

0

0

'o

E

L-

rO

'0

C

E

a,

V)

E

CP

V

C

E

0

0

i l

-C

E

E

o

(D

-o(_D <

U) -0

0 0

0

E E EO

I I I

0 c 0

0 0 0

(PX)

0

0

E

w

--

c(0

0o

< 7 < 7 < < ,o

:. / O / O 0

0 c / 0 0 V

IO

II II / II II

o0 J / / O

) / '' a,

c 4

0 C

U

09 '-

0 .,.4

'- O 4

-I

0

C o

0

*d o

U-

Sa,

C -0O

0 J 0 0

,-4

/ / 4J

O

0-

4 .J

% %o od o

( ) 2 CI

.0

cd

0 0

o

C-

0 0 0 0 0 0

(VW) ba,

0o

0

0

0

r)

-O

O

- N

C

3

o a

O LL

0

'O

C'j

cJ

'O

(VW) ba1

0

O

0

o

ro

0

0

0

I-

U0

E

E

o-

o

N

0

'0

m |

Eq. [5.18]. To check out this possibility, hfe, f, and hE were measured

independently and used to compute I according to Eq. [5.19]. The result-

eq

ing plots are shown in Fig. 4.14 along with the measured plots. The dis-

crepancy is so great that the conclusion is obvious. These FPT-100 units

have extra noise sources in addition to the 1/f type and the ones consider-

ed in this work. To show that this was the case, in a conclusive way, the

following experiment was carried out. Noise spectra were measured with

the device operating as a phototransistor and as a conventional transistor.

In the transistor mode, the base current was supplied through a 3 M f re-

sistor. The measured spectra are compared in Figs. 4.15 and 4.13 for the

Motorola and Fairchild devices respectively. The Motorola device used here

is not the unit of Fig. 4.9. For the Motorola unit, the low-frequency

plateau of the phototransistor spectrum is 3 db above the transistor plateau.

This is what is expected as explained in Chapter III in connection with Eq.

[3.13]. To repeat, the phototransistor is 3 db noisier because of the slot

noise contributed by (IBC + IpH). On the other hand, the Fairchild device

shows a noise reduction of less than 1 db at ICEO of 500 pA and practically

no reduction at 10 and 50 pA. This is a clear indication that the noise

mechanism responsible for the plateaus seen in this device is not associa-

ted with Eq. [5.18]. The behavior of the 1/f portion of the spectrum seems

to indicate that the extra noise has its origin at the surface. We notice

that in the 1/f region, the transistor mode of the Motorola device is about

2 db noisier than the phototransistor mode. This can be attributed to the

extra noise contributed by the base metal contact. However, this is not

observed in the 1/f spectrum of the Fairchild unit which leads to the con-

clusion that the surface noise of this unit is so strong that it overrides

the 1/f type of noise of its base contact. Without any knowledge of the

<

0 "

o o,

II 4- .

w oc

H 0.

o

N

r

C--

CC

o a

0

'0

1l O

S 0 o0 0 C

(VW) be,

62

fabrication process or physical parameters of this device, it is difficult

to draw any other more illuminating conclusion. Nevertheless, one practical

conclusion that can be drawn from these experiments is that this type of

two-fold noise measurement can be used to detect flaws in the device which

would go undetected through other types of measurements on the device.

University of Florida Phototransistor

The results obtained from the Fairchild FPT-100 seemed to justify the

effort of fabricating phototransistors whose processing and physical param-

eters would be known. Using the facilities of the Microelectronics Labora-

tory of the University of Florida, three-lead phototransistors were fabri-

cated. Appendix B gives the details of the process steps and the device

physical parameters.

The unit with the largest hFE was chosen for measurements. Figure

4.16 shows a plot of its spectra. The familiar shape of the T-I and

Motorola devices is recognized here. Figure 4.17 compares m2hFE, mhFE,

and hE with results similar to those of Figure 4.10. Figure 4.18 sho:s

m(IE) = hfe/hFE computed in the same way as in Figure 4.11. The agreement

is again good between values of m computed by independent approaches.

Figure 4.19 compares values of f obtained from the three independent

measurements used earlier. Agreement is again excellent. These measure-

ments on the University of Florida phototransistor reiterate the conclu-

sions already stated in the section on the Motorola device. In addition,

we can confidently add that no special tricks are needed in order to fab-

ricate a phototransistor with good noise performance like the T-I, Motorola,

and U. of F. units.

63

Sro

0

0

.,

C

&I

.0

-t

-oo

LtC

IL- II

0

/ -I

< EQ

UJ/ )) c .

ZU-

LOC-4

0 / 0

,-4

o 0 0

_0 0 0

^LUM) bo

0

0

0

o

C

0)

E

C E

Vj)

0

E- E

E 0 )

-C 0

E E

o a >0

to to ro o -

o 0 0 -o C

hO IC, C\j

UL LL LL.

C -c .C

E E

E

b

OJ

(

ro

IC

I

141

tT4

0 0

-4

4*-1 ,-

0

Ji ii

S c

Q C-

U

W 0

11 0

.

>

o-

0

0

r-

C 0

'-4 [1

cJ

< -

I C

0

0

70

w

I m

EE

E 4J,'

E o "-

CL E

o 0

a L

_ 1 a O

0 0 00H

L- >

> U_4

o 0 0 0

rO \ o o -

(ZH)Oi)

o <13

Computation of High-Frequency Effects

From the foregoing experimental results, it is apparent that the last

three terms of Eq. [5.26] are not sufficiently strong to be detected ex-

perimentally in the frequency range where the phototransistor is usetul.

Therefore, in order to determine their relative importance, they have been

evaluated numeiically with a co.rputer. The physical parameters of the U.

of F. device given in Appendix B have been used to compute the admittance

Y and Y The expressions for these admittances correspond to those of

e ce

a transistor with a uniformly doped base. The narrow electrical width of

the base of the U. of F. device mike this choice plausible. The expressions

used are

Y = (qlE/kT) + tn nh [ .8]

1/2

tanh (W/L )(l + jTr )

-h 1/2 -

h fe + jT n) sinh(V/L )

fe 1/2 jw9]

ce hfe I )(q E/kT) sinh (U/L )(1 + T )/29

The expression used to compute the current gain is

ce aCCT [4.10]

ye ye + jW(CCT + C ET)

Computations .cre carried out for several current levels with the

same result. The correction terms due to high-frequency effects are several

orders of magnitude below the main low-frequency term. Representative

results are tabulated in Table 4.1 for ICEO = .ImA.

CEO

TABLE 4.1

Comparison Between the Low- and High-Frequency Components

of the Output Noise

The net contribution is negative because of the term Re IB(g Y )

The product of the imaginary parts of 6 and Y makes this term negative.

ce

This product turned out to always be larger than the rest of the terr.ms

since these other terms are proportional to the difference between L.':o

very close together numbers; i.e., (ge ge) and (g Re(Yc))

Ce eo ceo ce)

I in mA.

eq

Frequency Low-frequency High-frequency

in KHz term term

10 107.5 -.681x10-4

50 52.7 -.742xio- 3

102 20.4 -.107x10"2

5x102 1.0 -.125xl0-2

105 .34 -.126x10-2

5.103 .11 -.126x10-2

10 .10 -.126x10"2

5xl04 .10 -.126x10"2

CHAPTER V

CONCLUSIONS

In the introduction of Chapter I it was stated that the main objective

of this investigation was to study the noise performance of phototransis-

tors to thus fill out a gap existing in the published literature on photo-

transistor characterization. This objective has been accomplished. In

doing so, another facet of phototransistor characterization has been

polished. That is, a numerical solution to compute the quantum efficiency

of the device has been developed which is superior to a previous analytical

expression.

An expression for the noise at the output of a phototransistor has

been derived for low frequencies along with terms to correct for high-

frequency effects. As experimental evidence and numerical calculations

show the high-frequency terms can be safely neglected.

Extensive noise, a.c., and d.c. measurements have been carried out

on several types of silicon phototransistors. The measured spectra agree

quantitatively with those predicted analytically. Excellent agreement be-

2

tween the values of hfe /i- and f obtained from noise measurements and the

te i-E B

other independent measurements provesconclusively the correctness of the

noise analysis. An inmrediate consequence of this agreement is that noise

measurements on phototransistors car. be used to characterize then in terms

of hfe/hF and f This is truly useful since in imaging arrays made up of

phototransistors, there are no base leads available. Hence, it is not idle

speculation to propose using noise measurements to diagnose malfunctioning

in sensing arrays. This is quite feasible since real-time spectrum analyzers,

like the G.R. system, have almost removed the time factor from noise

measurements.

The type of two-fold noise measurement carried on the Fairchild device

clearly indicates the wider scope of noise measurements. Although the

source of noise causing the distorted spectrum was not identified, it should

be recognized that the problem would have gone undetected through other types

of measurements on the device. Thus, as it was suggested in the introduc-

tion, one can gain more information about phototransistors using noise

*measurements because of their more revealing nature.

To extend the usefulness of the basic results obtained, an expression

for the NEP was derived. This expression predicts that the NEP is almost

constant up to frequencies near f at which the device has no useful gain.

For this frequency range, the NEP of a phototransistor equals that of the

collector-base photodiode multiplied by a factor of \2. Thus we can now

compare the two detectors by saying that while they practically have the

se-me detectivity, the phototransistor also has useful gain at the lower

frequencies.

The.expression for the NEP involves the quantum efficiency of the

collector-base photodiode. Since the expression derived by Gary and Lin-

vill for the base region was found lacking, a numerical solution of the

continuity equation was implemented to compute nPTD. The results from the

numerical solution indicate that the analytical expression is only useful

for low lifetime and high surface recombination velocity. This combination

of parameters is not found in good photodiodes. Hence, it is desired to

predict the quantum efficiency of good photodiodes, the numerical approach

must be used.

Extension.of this work might involve experimental verification of the

results presented here on the NEP and n,,_.

APPENDIX A

NUIEERICAL SOLUTION OF THE CONTINUITY EQUATION

It is desired to solve Eq. [2.16] subject to the boundary conditions

6ivu i by Eqs. [2.9) anu [2.10]. For convenience Eq. [2.16] is repeated

here

d2n IdD D dC dn

D + ---

n 2 dx C dx dx

n D - n =

C dx dx d C dx 1

F(v)a exp(-ox) [A.1]

This equation is noraalized using

x [A.2]

X -

D (x)

Dn

7 D B) [A.3]

dn

n dv dX .

v = [A.4]

F(v)dX F(vB)

Dn(x B Dn(xB)

Notice that dv/dX evaluated at x gives the desired T "

The expression concocted byCaughey and Thonas has been used to ex-

press the dependence of diffusivity on impurity concentration or x

52.6

D (x) = 5 + 1.68

1. + C( 1

8.5x:1015

Using the normalized variables, Eq. [A.I] becomes

dr2 d _7 dC dv

7 + +2

dd

7 2 + X + C 'dX dX

x [A.6]

2

SdC _d I dC X"B

C dX dX d- C dX C d D (X )

= xa exp(-ox)

This equation is next reduced to two simultaneous equations of first degree

in the normalized variables v and z = dv/dX

dz I d- __ dC

dX 7 dX C dX [A.7]

2

1 I dC +d 7 d 1 dC x B

7 C dXX d d): C dX D(X)

n n B

x a e:xp(-Cz)

7

dv [A.8]

To simplify this, let A, B, and C be the coefficients of Eq. [A.7]

dz

= A.z B.v Cexp(-:.) [A.9]

Eqs. [A.8] ai:d [A.9] are subject to the boundary conditions [2.9] and

[2.10]. In order to integrate these equations, one needs to know v(0) and

z(0). Boundary condition [2.9] gives z(0) in terms of v(0)

z(0) =- D- .v(0) [A.10]

o n = 0

The scheme used to satisfy both boundary conditions consists of choosing

an initial v.(0) and thereby z.(0), integrate Eq. [A.8] and [A.9] up to

X = XB and check if v(XB) = 0 as boundary condition [2.10] demands. If

v(XB) ? 0, a new value of vi(0) must be chosen. The garden-hose method 1

has been used in order to correct the initial choice v.(0). It is obvious

that v(XB) is some unknown function of v(0) as shown in Figure A.1

v(X)

S((0)

---- *V(0)

v.(0) v. (0)

Figure A.I. Hypothesized dependence of v(XB) on v(0)

We want to choose the value of v(0) which is a root of v(XB). It is

well known that such root can be found using Newton's method. The garden-

hose method is an adaptation of Newton's method to the system of Eqs. [A.8]

and [A.9].

For simplicity's sake, let v.(0) = M and differentiate Eqs. [A.8]

and [A.9] with respect to M and reverse the order of the various deriva-

tives

S( ) 6VM 6[A.11]

S( z ) =-A. B. [A.12]

Let u = 6v/6M and y = 6z/&4, then we have the system

u [A. 15]

A.y B.u [A.14]

which is subject to boundary conditions easily obtained from Eq. [A.10]

u(0) = () = [A. 15

D, -& (o)

y(O) Sx 1 u(O) [A.16]

D --C dX

n = 0

Notice that for this second system both boundary conditions are at X = 0

and are known. Then for an arbitrary choice of v.(0), this last, so to

speak, auxiliary system is solved simultaneously with the m3in system. The

solution of u at X = XB gives us the rate of change of v(XB) with respect

to v(0) evaluated at v.(0) as shown in Figure A.I. Using this slope, a

new improved v.(0) can be obtained from

v(XB)

v!(O) = v(0) ) [A.17]

v vi(O) u(XB)

v(0) = vi(0)

The cycle is repeated until v(XB) is as close to zero as wished.

16

In order to integrate these equations, a simple predictor-corrector

has been used. To use this predictor-corrector, the values of v, dv/dX,

z, and dz/dX, and also of u, du/dX, y, and dy/dX must be known at the

first two points of the integration net. It has already been explained

how to obtain v, z, u, and y at the first point X = 0. Using these

values in Eqs. [A.8], [A.9], [A.1I] and [A.14] gives the correspond-

ing derivatives. At the second point, the values have been obtained by

means of Taylor's series expansion about the first point. The series

have been truncated after the third power term.

A listing of the FORTRAN program starts on the next page.

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APPENDIX B

PHOTOTRANSISTOR FABRICATION

The purpose of this appendix is to present the main steps followed

during the fabrication of phototransistors using the facilities of the

Microelectronics Laboratory of the University of Florida.

A good phototransistor must have high quantum efficiency. From the

results presented in Chapter II, this calls for low surface recombination

velocity and high minority carrier lifetime in the base region. As a

rule of thumb, these objectives are achieved by keeping the wafer free

from foreign contaminants. This was achieved by repeatedly cleaning the

surface of the wafer at different steps throughout the process and by

gettering the contaminants from the oxide after the device had been made.

Next, some of the main steps are described. A N-lll Silicon wafer of 1

ohm-cm resistivity was used with a conventional circular geometry.

Fabrication Steps

Wafer Cleaning

The wafer was cleaned by successive ultrasonic baths in trychloro-

ethylene and acetone. A D-I water rinse was used to remove these agents

after each bath. Next, the wafer was dried by spinning it on a vacuum

chuck. Spraying with the nitrogen hose was avoided since the nitrogen

could be contaminated. A last bath in boiling NHO was used to remove

metal ions. This was 5 minutes long. Rinsing and spinning followed.

Oxida ion

The wafer was put in the oxidation tube at 11500C. The first 15

minutes were in a wet oxygen atmosphere. The flow rate of oxygen was set

at 0.75 cc/min. The oxygen was stopped and a nitrogen flow at 0.8 cc/min

was maintained for 5 minutes. A slow pull, 5 minutes long, was used to

remove the wafer from the tube. After cooling,an HF dip was done to re-

move any contaminants that may be on the oxide. The resulting oxide layer

was 0.5p.

Base Diffusion

After oxidation, steps involving masking, developing, etching, and

photoresist removal were carried out using conventional techniques. The

oxide thickness was reduced to 0.12i after all these steps.

The tube was predoped for 1 hour before actual prcdeposition. This was

done to expulse foreign gases from the tube and insure a uniform doping

environ.r,'nnt inside the tube. Th2 boron source was a .IxLurc oL dULaULIU

and argon, 1000 ppm strong. The boron flew was set at 75 cc/min and that

of the carrier gases nitrogen and oxygen at 0.4 cc/min and .025 cc/min

respectively.

After quick dip of 2 seconds in HF, the wafer was loaded in the tube

and predoped for 12.5 minutes at 9900C. Under these conditions, a V/I of

25 f was obtained. An HF bath was then used to remove the B 0 that had

formed during the predeposition. The whole wafer was then sealed by grow-

ing SiO at 11000C for 20 minutes in a wet oxygen flow of 0.75 cc/min.

The oxide thickness over the base was 0.53 and over the collector was 0.5.-

The wafer was then loaded back into the tube for a drive-in which lasted

2 hours at 1150C in a nitrogen flow of 1.2 cc/min. The final V/I was 65 Q.

Emitter Diffusion

The steps necessary to open the emitter window were carried through

observing strict cleanliness. A dip followed to obtain a uniform oxide

thickness over the whole wafer. The oxide was thinned to .08[ which might

have not been thick enough to be a good protective barrier. However, the

process was continued and no problems were encountered.

The phosphorus source was a mixture of phosphine gas and argon. It

was 10,000 ppm. strong. The tube was predoped for 15 minutes with the

source at 1.5 cc/min, and the nitrogen and the oxygen both at .1 cc/min.

With these flow rates, the wafer was loaded for a 15-minute predeposition

at 11000C after which it was quickly removed from the tube. The measured

V/I was .56 f2. A phosphorus drive-in followed at 10500C for 8 minutes in

a wet oxygen flow of .1 cc/min and an additional 3 minutes in a dry oxygen

atmosphere. The final V/I was .50 f2, and the oxide thicknesses were

.12L over the emitter, .21p over the base, and .28u elsewhere.

Gettering of Metal Precipitates

In order to remove mobile-charged metal ions from the device, a glassy

17

layer of phosphorus oxide silicon oxide has been used as a getter.7 Thus,

after the emitter diffusion, another phosphorus predeposition and drive-in

steps were performed. The amount of predeposited phosphorus and duration

of the drive-in cycle were determined by the oxide thicknesses obtained

after the emitter diffusion. Thus, the predeposition lasted only 3 minutes

with the phosphorus flow rate set at 1.5 cc/min, the nitrogen at .1 cc/min,

and the oxygen at .075 cc/min. The tube temperature was 10000C. The

source was shut off to proceed with the drive-in cycle. During the drive-

in cycle, the wafer stayed for 10 minutes inside the tube in an atmosphere

of dry oxygen at 10000C. The oxygen flow-rate was .2 cc/min. Then, the

oxygen was shut off while simultaneously setting the nitrogen flow at

.055 cc/min in preparation for an 8-minute slow-pull of the wafer from

the tube.

Having finished the double diffusion sequence, conventional techniques

were used to make ohmic contacts using aluminum. After the contacts were

made, the wafer was scribed and broken, and the individual devices mounted

on TO-5 cons with leads bonded to them. The following parameters were

measured using conventional techniques.

TABLE B.1

Physical Parameters of the U. of F. Phototransistor

Collector impurity concentration

Impurity concentration at the surface of the base

Metallurgical collector-base junction depth

Metallurgical emitter-base junction depth

Base radius

Emitter radius

5xl015 cm-

9x1017 cm-

-4

5.52x10 cm

-h

2.28x10 cri

-2

1.69x10O2 cm

6.55x10- cm