• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Figures
 List of Figures
 Abstract
 Introduction
 Band structure and effective...
 Mobility and scattering relaxation...
 Hole density and resistivity
 The half factor in p-type...
 Experimental procedures
 Comparison of theoretical and experimental...
 Summary and conclusions
 Appendix
 Reference
 Biographical sketch
 Copyright














Group Title: mobility, resistivity and carrier density in p-type silicon doped with boron, gallium and indium
Title: The mobility, resistivity and carrier density in p-type silicon doped with boron, gallium and indium
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082458/00001
 Material Information
Title: The mobility, resistivity and carrier density in p-type silicon doped with boron, gallium and indium
Physical Description: xvii, 148 leaves : ill. ; 28 cm.
Language: English
Creator: Linares, Luis Carlos, 1944-
Publication Date: 1979
 Subjects
Subject: Semiconductor doping   ( lcsh )
Silicon   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1979.
Bibliography: Bibliography: leaves 144-147.
Statement of Responsibility: by Luis Carlos Linares.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082458
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000099739
oclc - 07119890
notis - AAL5197

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
        Page x
    List of Figures
        Page xi
        Page xii
        Page xiii
        Page xiv
    Abstract
        Page xv
        Page xvi
        Page xvii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Band structure and effective mass
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
    Mobility and scattering relaxation time
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
    Hole density and resistivity
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
    The half factor in p-type silicon
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
    Experimental procedures
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
    Comparison of theoretical and experimental results
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
    Summary and conclusions
        Page 103
        Page 104
        Page 105
        Page 106
    Appendix
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
    Reference
        Page 143
        Page 144
        Page 145
        Page 146
    Biographical sketch
        Page 147
        Page 148
    Copyright
        Copyright
Full Text








THE MOBILITY, RESISTIVITY AND CARRIER DENSITY
IN p-TYPE SILICON DOPED WITH BORON, GALLIUM AND INDIUM











By

LUIS CARLOS LINARES


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


1979





























TO MY FAMILY

FOR THEIR PATIENT SUPPORT














ACKNOWLEDGMENTS


I gratefully express my appreciation to the members of my super-

visory committee for their support and cooperation. In particular,

I thank Dr. S. S. Li for his guidance, enthusiasm and professional exam-

ple, and Dr. F. A. Lindholm for his continuing interest in the research.

A special expression of appreciation is due Dr. A. D. Sutherland for

his advice and support during the first year of my graduate work.

I am also indebted to D. Yuen for his help with measurements on

the gallium- and indium-doped samples, to M. Riley for the fabrication

and measurement of the boron samples, and to W. Axson and R. Wilfinger

for their invaluable help with various laboratory procedures.

This investigation was made possible by the Air Force Institute of

Technology. The research was jointly supported by the National Bureau

of Standards Contract No. 7-35741 and the National Science Foundation

Grant No. ENG 76-81828.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . . . . . .

TABLE OF CONTENTS . . . . . .

LIST OF FIGURES . . . ... .

KEY TO SYMBOLS . . . . . . .

ABSTRACT . . . . . . . .

CHAPTER


I INTRODUCTION . . . . . . . . .

II BAND STRUCTURE AND EFFECTIVE MASS . . .

2.1 Introduction . . . . . . .

2.2 The Valence Band Structure of Silicon .

2.3 Effective Mass Formulation .

2.4 Discussion . . . . . . . .

III MOBILITY AND SCATTERING RELAXATION TIME .

3.1 Introduction . . . . . . .

3.2 Mobility and Average Scattering Relaxation

3.3 Acoustical Phonon Scattering . . .

3.4 Optical Phonon Scattering . . . .

3.5 Ionized Impurity Scattering . . .. .

3.6 Neutral Impurity Scattering .. . .

3.7 Effect of Hole-Hole Scattering. . .

3.8 Mobility in the Combined Valence Band .


. .

. .

. .

. .

. .

. .

. .*

. .

Time

. .

. .

. .


PAGE

iii

iv

vii

xi

xv



* 1

S 6

6

8

S12

S26

29

29

30

33

34

35

36

37









PAGE

IV HOLE DENSITY AND RESISTIVITY . . . . . . . 49

4.1 Introduction . . . . . . . . . . 49

4.2 Ionization of Impurity Atoms . . . .. . . 49

4.3 Resistivity of p-Type Silicon . . . . . 55

V THE HALL FACTOR IN p-TYPE SILICON . . . . ... 62

5.1 Introduction . . .. . . . . ..... .. 62

5.2 The Hall Factor . . . . . .... . 63

5.3 The Mass Anisotropy Factor ....... . . .. 66

5.4 The Scattering Factor . . . . . . .... 67

5.5 Hall Mobility and Hall Factor in the
Combined Valence Band ... . . . . . . 72

VI EXPERIMENTAL PROCEDURES . . . . . . ... .78

6.1 Introduction . . . . . . . . . . 78

6.2 Fabrication Procedure ... ..... ..... .79

6.3 Experimental Measurements . . . . . . 80

'VII COMPARISON OF THEORETICAL AND
EXPERIMENTAL RESULTS . . . . ...... . . .. 84

7.1 Conductivity Mobility . . . . . ... . . 84

7.2 Resistivity . . . . . . . . . . 86

7.3 Hall Mobility . . ... . . . . ... . 93

7.4 Hall Factor . . . . . ... . . 93

VIII SUMMARY AND CONCLUSIONS . . . . . . .... 103

APPENDIX

A FABRICATION PROCEDURE AND TEST STRUCTURES .. . . 107

B EXPERIMENTAL SETUP AND DATA . . . . . . .. 119

C COMPUTER PROGRAM ....... . . . . . 127









PAGE

REFERENCES ........................... 144

BIOGRAPHICAL SKETCH . . . . . ... . . . . . . ....148
















LIST OF FIGURES


FIGURE

2.1 Simplified valence band structure of silicon . . .

2.2 Temperature dependence of the density-of-state
effective masses . . . . . . . .

2.3 Temperature dependence of the conductivity
effective masses . . . . . . . . .

2.4 Temperature dependence of the Hall effective masses

2.5 The acceptor density dependence of the combined
conductivity effective mass of holes in silicon
as a function of temperature . . . . . . .

2.6 The acceptor density dependence of the combined Hall
effective mass of holes in silicon as a function
of temperature . . . . . . . . . .

3.1 The calculated hole mobility vs dopant density for
boron-doped silicon with temperature as a parameter

3.2 The calculated hole mobility vs dopant density for
gallium-doped silicon with temperature as a parameter

3,3 The calculated hole mobility vs dopant density for
indium-doped silicon with temperature as a parameter

3.4 The calculated hole mobility vs temperature for
boron-doped silicon with dopant density as a parameter

3.5 The calculated hole mobility vs temperature for
gallium-doped silicon with dopant density as a
parameter . . . . . . . . . .

3.6, The calculated hole mobility vs temperature for
indium-doped silicon with dopant density as a
parameter . . . . . . . . ... . .


PAGE

. 11


. . 20


. . 22

. . 23


. . 25


. . 42


. . 43


. . 44


. .


. .








FIGURE PAGE

4.1 Theoretical calculations of the ratio of ionized
and total boron density vs.boron density with
temperature as a parameter ... . . . . 52

4.2 Theoretical calculations of the ratio of ionized
and total gallium density vs gallium density with
temperature as a parameter . . . . . . ... 53

4.3 Theoretical calculations of the ratio of ionized
and total indium density vs indium density with
temperature as a parameter . . . . . . ... 54

4.4 Theoretical calculations of resistivity vs temperature
for boron-doped silicon with dopant density as a
parameter. . . . . . . . .. .. . 56

4.5 Theoretical calculations of resistivity vs temperature
for gallium-doped silicon with dopant density as a
parameter . . . . . . . . . . . . 57

4.6 Theoretical calculations of resistivity vs temperature
for indium-doped silicon with dopant density as a
parameter . . . . . . . . .. .. .. .. .58

4.7 Theoretical calculations of resistivity vs dopant
density for boron-doped silicon with temperature
as a parameter . . . . . . . ... .... 59

4.8 Theoretical calculations of resistivity vs dopant
density for gallium-doped silicon with temperature
as a parameter . . . . . . . .... ... .. 60

4.9 Theoretical calculations of resistivity vs dopant
density for indium-doped silicon with temperature
as a parameter . . . ... . . . . ..... 61

5.1 The mass anisotropy factor rA as a function of temperature
for various impurity dopant densities . . . . . 68

5.2 The mass anisotropy factor rA as a function of impurity
dopant density for various temperatures . . .... ... 69

5.3 The scattering factor rS as a function of temperature
for boron-doped silicon with dopant density as a
parameter . . . . . . . . ... .. .. . .70

5.4 The scattering factor rS.as a function of dopant
density for boron-doped silicon with temperature
as a parameter .. . . . . . . . . .. .71


viii









FIGURE


5.5 Theoretical Hall factor vs temperature for
boron-doped silicon with dopant density as
a parameter . .. . . . . . . . .

5.6 Theoretical Hall factor vs dopant density for
boron-doped silicon with temperature as a parameter

5.7 Theoretical Hall mobility as a function of temperature
for boron-doped silicon with dopant density as a


parameter . .


5.8 Theoretical Hall mobility as a function of dopant
density for boron-doped silicon with temperature
as a parameter . . . . . . . . .

7.1 Hole mobility vs hole density for boron-doped
silicon at 300 K . . . . . . . . .

7.2 Resistivity vs dopant density for boron-doped
silicon at 300 K . . . . . . . . .

7.3 Resistivity vs dopant density for gallium- and
indium-doped silicon at 300 K . . . . . .

7.4 Resistivity vs temperature for the.boron-doped
silicon samples . . . . . . . . .

7.5 Resistivity vs temperature for the gallium-doped
silicon samples . . . . . . .

7.6 Resistivity vs temperature for the indium-doped
silicon samples . . . . . . . . .


. . . . . . . . . . 76


S77


. 85


. 87


S88


90


. 91


92


7.7 Hall
NA

7.8 Hall
NA =

7.9 Hall
NA =

7.10 Hall

NA =
7.11 Hall
NA =


mobility vs temperature for gallium-doped sample.
4.25x1015 cm 3 . . . . . . . . .

mobility vs temperature for gallium-doped sample.
4.09x1016 cm 3 . . . . . . . .

mobility vs temperature for gallium-doped sample.
1.26x1017 c-3
1.26x 01 cm . . . . . . . . .

mobility vs temperature for gallium-doped sample.
3.46x1017 cm-3

mobility vs temperature for indium-doped sample.
16 -3
4.64x10 cm-


ix


PAGE


. .








FIGURE PAGE

7.12 Hall mobility vs temperature for indium-doped sample.
NA = 6.44x1016 cm-3 . ................ .99

7.13 Hall factor vs dopant density for p-type silicon
at 300 K . . . . . . . . . . ... . .. 100














KEY TO SYMBOLS


Inverse mass band parameter

Area of the base-collector diode

Defformation potential constant (acoustic phonon scattering)

Inverse mass band parameter

Defformation potential constant (optical phonon scattering)

Inverse mass band parameter

Longitudinal sound velocity in silicon

Transverse sound velocity in silicon

Energy of holes

Magnitude of the electronic charge

Acceptor impurity energy level

Fermi energy level

Valence band edge

Binding energy of neutral acceptors

The Fermi-Dirac function

Fermi-Dirac integral of order 1/2

Ground state degeneracy

Plank's constant divided by 2'

Current

Current density

Wave vector

Boltzmann's constant










m*

C
m2*
mi
m*

m*
mDi
m*
G

H
m*




NN

n











rHi
RH
Hi'
rHi

rSi
S

T

V

VH


Heavy-hole mass at 4.2 K

Light-hole mass at 4.2 K

Conductivity effective mass in the combined band

Conductivity effective mass in band i

Density-of-state effective mass in the combined band

Density-of-state effective mass in band i

Geometric mean mass

Hall effective mass in the combined band

Hall effective mass in band i

Total acceptor impurity density

Ionized acceptor impurity density

Neutral impurity density

Phonon distribution function

Effective.density of valence band states

Hole density in band i

Effective screening hole density

Mass anisotropy factor in band i

Hall coefficient in the combined band

The Hall factor in the combined band

Hall coefficient-in band i

The Hall factor in band i

Scattering factor in band i

Probe spacing

Absolute temperature

Voltage

Hall voltage








w Thickness of the chip

3 Ratio of defformation potential constants

Y Function of band mass parameters

Yi. Ratio of density-of-state effective masses

Yhh Hole-hole reduction factor for acoustic phonon scattering

Yhh Hole-hole reduction factor for ionized impurity scattering

Yhh Hole-hole reduction factor for optical phonon scattering

A Energy of spin orbit splitting

E Reduced energy (E/k T)

eI Variable of integration

E2 Variable of integration

Es Relative dielectric constant

r Limit of integration

S. Reduced Fermi-energy

nI Scaling factor
6 Spherical coordinate

OD Debye temperature

pC Conductivity mobility in the combined band

.Ci Conductivity mobility in band i
pH Hall mobility in the combined band
Limit of integration defined in Figure 2.1

p Resistivity of holes

Ps Density of silicon

OC Electrical conductivity
OH Hall conductivity

T Total scattering relaxation time

Tac Acoustic phonon scattering relaxation time in band i
ac v1


xiii








Tli Ionized impurity scattering relaxation time in band i

Tij Total interband scattering relaxation time

Tii Total intraband scattering relaxation time

i Neutral impurity scattering relaxation time in band i
Toi Optical phonon scattering relaxation time in band i

S Adjustable scattering constant
Spherical coordinate














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


THE MOBILITY, RESISTIVITY AND CARRIER DENSITY
IN p-TYPE SILICON DOPED WITH BORON, GALLIUM AND INDIUM

By

Luis Carlos Linares

August 1979
Chairman: Sheng-San Li
Major Department: Electrical Engineering

Using the relaxation time approximation and a three-band model

(i.e., nonparabolic light-hole band, parabolic heavy-hole and split-off

bands), a derivation involving the use of the Boltzmann transport theory

was applied to obtain expressions for the valence band density-of-states

effective mass, m*, the valence band conductivity effective mass, mC,

and the valence band Hall effective mass, m*, of holes in p-type sili-

con. Values of.effective mass calculated from this model reveal the

temperature and dopant density dependence of the effective mass due to

the nonparabolic shape of the bands. With these values of effective

mass and the three-band model, theoretical calculations of hole mobil-

ity, resistivity, Hall factor and Hall mobility were conducted for

p-type silicon doped with boron, gallium and indium for dopant densities

from 101 to 101 cm3 and temperatures between 100 and 400 K. Scatter-

ing contributions from acoustical and optical phonons, and ionized and

neutral impurities were considered. In addition, interband scattering








for the case of acoustical phonons, the effect of hole-hole scattering

on the various scattering mechanisms, and the nonparabolicity of the

valence band were also taken into account in the calculations. The

valence band density-of-states effective mass, mD, was found to vary

from 0.6567 mo at 100 K to 0.8265 mo at 400 K, while the valence band

conductivity effective mass, m*, increased from a value of 0.3604 m

at 100 K to a value of 0.4910 m at 400 K. The valence band Hall effec-

tive mass, mH, varies from 0.2850 mo at 100 K to 0.5273 m at 400 K.

The masses mD and m~ showed little change with dopant density, but

mH varied by as much, as 63 percent at 100 K over the range of dopant

densities considered. The Hall factor was expressed in terms of a mass

anisotropy factor and a scattering factor. These two components of the

Hall factor were separately evaluated to emphasize their individual

contributions to the Hall factor. Theoretical values of the Hall factor

vary between 1.73 and 0.77 over the dopant density and temperature

ranges.

To verify our theoretical calculations, resistivity and Hall

coefficient measurements were performed on silicon samples doped with

boron, gallium and indium with dopant densities ranging from 4.25x1015

to 9.07x1017 cm-3, for 100 T < 400 K, using planar square array test

structures. Agreement between the calculated and measured resistivity

values was within 10 percent over the range of temperatures and dopant

density studied. Agreement between our calculated and experimental

values of Hall mobility was within 15 percent for temperatures greater

than 150 K. At 300 K, agreement between theoretical values of Hall

factor and those deduced from experimental data of this work and that

of other studies was within 15 percent for dopant densities greater








16 -3
than 5x10 cm From the results of this study, we conclude that the

theoretical model and expressions developed here can accurately describe

the mobility and resistivity in p-type silicon. The omission of band

anisotropy considerations, however, introduces considerable error in the

evaluation of the Hall factor for low dopant densities.


xvii














CHAPTER I

INTRODUCTION


The goal of this study has been to measure and compare with theory

the resistivity and Hall mobility of holes in silicon doped with gallium

and indium as functions of temperature and dopant density. Data taken

on boron-doped silicon were also included in order to further confirm the

adequacy of the theoretical model. In order to determine theoretically

the resistivity and Hall mobility, one must first calculate the conduc-

tivity mobility, the density of holes, and the Hall factor. This can

only be done with a thorough understanding of the energy band structure,

the scattering mechanisms involved, and the carrier statistics.

The application of an electric or a magnetic field to a crystal

results in a variety of carrier transport phenomena. These phenomena

are associated with the motion of current carriers in the conduction or

valence bands. The free charge carriers in a semiconductor will acquire

a drift velocity under the influence of an applied electric field. This

velocity is the net result of the momentum gained from the externally

applied field, and the momentum lost in collisions which tend to randomize

the carrier momentum [1]. If the field is expressed in volts per centi-

meter, and the velocity in centimeters per second, a mobility is defined

as the incremental average speed per unit electric field, and is expressed

in squared centimeters per volt second. The velocity, and consequently

the mobility, is determinedby the different types of collisions which

the carriers undergo. Collisions of carriers with lattice atoms which

1








are out of their equilibrium positions because of thermal vibration,

provide an upper limit to the mobility. Scattering of the carriers is

also caused by impurities, both ionized and neutral. At high tempera-

tures and low dopant densities, scattering by lattice phonons is more

effective while at low temperatures and high impurity densities, scat-

tering by ionized and neutral impurities predominates. In addition,

the effects of hole-hole scattering on the lattice and ionized impurity

scattering mechanisms need to be considered. Thus in calculating the

mobility over a wide range of temperatures and dopant densities, all the

different scattering mechanisms must be taken into account. Besides the

mobility, the density of holes enters the problem of determining the

electrical resistivity. The density of holes associated with dopant

atoms is a function of the ionization energy of the dopant atom, the

temperature, and the degeneracy factor. Calculation of the Hall mobility

requires knowledge of the Hall factor which is a function of the scat-

tering mechanisms and effective masses.

For purposes of device design it is necessary to know the correct

relationship between the resistivity and dopant density at different

temperatures. Evaluation of the characteristics of semiconductor devices

and the study of transport phenomena in semiconductors requires an

accurate knowledge of variations in the effective mass, mobility, and

carrier density with changes in temperature and dopant density. Because

of this, numerous studies of mobility, resistivity, and hole density in

p-type silicon have been conducted [1-16]. However, due to the complexity

of the valence band of silicon and the various scattering mechanisms

involved, these studies, for the most part, have either been conducted

in temperature and dopant density ranges designed to explore only a








particular type of scattering mechanism, or have not advanced the theory

necessary to describe the experimental result. For example, Costato and

Reggiani [4] calculated the mobility of holes for pure p-type silicon in

which lattice scattering dominates; Braggins [1] considered nonparaboli-

city and all the relevant scattering mechanisms with the exception of

hole-hole scattering, but he limited his investigation to dopant densities

below 5x1016 cm-3 and low temperatures; Morin and Maita [5] considered

wide ranges of temperature and dopant densities, but did not provide a

theoretical examination of the data. Recently, Li [17] developed a

theoretical model capable of describing the mobility and resistivity of

p-type silicon over a wide range of temperatures and dopant densities.

This improved model was applied to the case of boron-doped silicon with

great success [17]. The improvement in the theory consisted mainly of

the inclusion of hole-hole scattering effects, and consideration of the

nonparabolic nature of the bands. In this study, Li's model [17] has

been improved by including consideration of interband scattering effects

on the acoustic phonon scattering mechanism, and has been applied to the

study of silicon doped with impurities other than boron.

With some exceptions [14-16], most of the research in p-type silicon

has been conducted with.boron as the doping impurity, since boron is the

shallowest acceptor in silicon and this material is widely available.

A very limited amount of data is available on silicon doped with deeper

impurities such as gallium and indium. These dopants, especially indium,

are of great interest to modern technology because of their application

to photo-detector devices. Curves of resistivity and mobility as func-

tions of dopant density [2,3] have been applied to characterizing

boron-doped starting material and diffused boron layers in silicon, and








have been found highly useful. Similar curves developed in this research

may be expected to be equally useful for characterizing and integrating

infrared detectors based on the deeper levels of indium andgallium with

on-chip silicon electronics. Application of a more complete theory of

mobility and resistivity [17] to the case of silicon doped with gallium

and indium should provide an accurate description of the transport of

holes in this material. These results may be of significant use in the

study and design of infrared photo-detector devices.

In this research the mobility, resistivity, and hole density have

been studied over a temperature range from 100 to 400 K and dopant

densities from 4.25x1015 to 9.05x101 cm3. Because of the complexity

brought about by heavy doping effects and uncertainties in accounting

for hole density and impurity density at high dopant densities, the

theoretical analysis has been restricted to densities below 1018 cm3 in

which the use of Boltzmann statistics is justified. The nonparabolic

nature of the valence-band structure and derivation of expressions for

the-temperature dependent effective masses are presented in Chapter II.

Since effective mass is directly related to the shape of the valence

bands, the result is an effective mass which varies with temperature and

dopant density. The mobility formulation includes consideration of the.

relevant scattering mechanisms and how these are modified by hole-hole

scattering effects. These scattering mechanisms are considered in detail

in Chapter III. Since the different scattering mechanisms which contri-

bute to the mobility have different temperature and energy dependence,

the use of numerical methods'and curve fitting has been applied in

analyzing the data. The temperature and dopant density dependence of

resistivity and hole density is analyzed in Chapter IV. In Chapter V,






5


the Hall factor is discussed, and theoretical calculations of Hall

mobility are presented. Fabrication techniques and experimental proce-

dures are described in Chapter VI. Comparisons of experimental results

with predictions based on the theory of Chapters III through V are made

in Chapter VII; in this chapter the theoretical results are also compared

with data published by other workers. Chapter VIII summarizes the

research and states the main conclusions derived from this work.














CHAPTER II

BAND STRUCTURE AND EFFECTIVE MASS


2.1 Introduction

The interpretation of transport properties in silicon and the model-

ing.of silicon junction devices depend on an accurate knowledge of values

of effective mass. The complex valence band structure of silicon leads

to difficulties in the study of transport properties of holes in this

material. Thus the development of a model incorporating the nonparabolic

nature of the band into a single parameter, the combined hole effective

mass, would greatly simplify the study of mobility, resistivity, and the

Hall effect in silicon. Including the band nonparabolicity in calcula-

tions of relaxation time via the effective mass formulation is a reason-

able procedure and has been applied effectively by Radcliffe [18] to

study acoustic phonon scattering, and by Barrie [19] to study optical

phonon and impurity scattering in nonparabolic bands. In this chapter

we will derive such a theoretical model for hole effective mass calcula-

tions in silicon.

Lax and Mavroides [20] have derived expressions for density-of-states

effective masses m*D and m*2 for the heavy-hole band and the light-hole

band, respectively, which lead to the generally accepted and quoted

value, mD = 0.591 m This value, however, can only be considered

applicable at 4.2 K, where m* = 0.537 m and m* = 0.153 m A number

of experimental data have been published which indicate both electron

and hole effective mass to be dependent both on.temperature and dopant
6








density [21,22]. Below 50 K, hole effective mass remains constant as

indicated in high frequency magnetoconductivity experiments [23].

However, at higher temperatures and for higher acceptor impurity densi-

ties, two mechanisms are responsible for the temperature dependence of

the effective mass: the thermal expansion of the lattice, and the

explicit effect of temperature. The effect of the thermal expansion

can be estimated from the stress dependence of the effective mass [24],

and has been shown to be negligible [21,25]. The explicit temperature

effect however is of great importance. It consists of three parts:

(a) the temperature variation of the Fermi distribution function in a

nonparabolic band, (b) the temperature dependent distribution function

of the split-off band, and (c) the temperature variation of the curva-

ture at the band extremum due to the interaction between holes and

lattice phonons.

Following the work of Lax and Mavroides [20], but using Fermi-Dirac

statistics.and a simplified model of the valence band structure for

silicon, Barber [25] obtained an expression for the density-of-states

effective mass, m*, which is temperature and hole-density dependent.

Barber, however, did not apply the nonparabolic model of the valence

band to the study of conductivity or Hall effective mass in p-type

silicon. Costato and Reggiani [26] also developed expressions for mD

and m*, the band conductivity effective mass, which show a variation

with temperature, but they neglected the effects of the split-off band

and the temperature variation of the band curvature.

In this study, the expressions for density-of-states effective mass,

conductivity effective mass, and Hall effective mass of holes are derived

based on the following definitions. The density-of-states effective








mass, m*, enters in the normalization of the distribution function; the

conductivity effective mass, m*, is the mass of a mobile charge carrier

under the influence of an external electric field; and the Hall effec-

tive mass, mH, is the mass of a mobile charge carrier under the applica-

tion of external electric and magnetic fields. The reason for these

particular definitions of effective masses is that the primary applica-

tion of this work is to generate improved theoretical calculations of

Hall mobility, resistivity, and conductivity mobility [17]. The

derived expressions were used to calculate hole effective masses in

p-type silicon over a wide range of temperature and dopant density.

Since the crystal structure of silicon has cubic symmetry, the ohmic

mobility and the low-field Hall coefficient are isotropic. An angular

average of the effective masses may be performed taking into account

separately the warping of the individual bands so that expressions for

mD, m, and m* of isotropic form can be derived. Values calculated from
these expressions differ from one another because of the warping and

nonparabolicity, and consequently effective mass in each band depends on

temperature and dopant density in its own way. The valence band struc-

ture of silicon is presented in Section 2.2, and in Section 2.3 expres-

sions for m*, mC, and m* are derived.


2.2 The Valence Band Structure of Silicon

Theoretical calculations by Kane [27] have established some basic

features of the valence band of silicon. It consists of heavy-hole and

light-hole bands, degenerate at k = 0, and a third band displaced down

in energy at T = 0 by spin orbit coupling.

The heavy-hole band is characterized by holes with an energy inde-

pendent, but direction-dependent effective mass. The light-hole band








is characterized by holes with an energy and direction-dependent effec-

tive mass. These two bands can be described by the E vs k relationship

[28]


__(2 +2 (k2k2 k2k2 k2k2
E(k) = 2-o Ak2 [B k4 + x yC(k +k k + k k )] (2.1)


where A, B, and C are the experimentally determined inverse mass band
2 2 2
parameters, k (k + ky + k) and the upper sign is associated with
x y z
the holes in the light-hole band, while the lower sign is associated

with the holes in the heavy-hole band. Values of A, B, and C are

obtained by cyclotron resonance measurements at 4 K [22,29].

Although warped, the bands are parabolic for small values of k.

However, for larger values of k, the bands become nonparabolic, and along

the <100> and <111> directions the heavy- and light-hole bands are

parallel over most of the Brillouin zone. This situation, however, is

not strictly valid for general directions [30]. The assumption of

overall parallelism, while questionable in III-V compounds, is reasonable

in the case of Ge and Si [27,31]. The split-off band is separated at

k = 0 by an energy A = 0.044 eV. [32], and is characterized by an effec-

tive mass which is independent of energy and direction. If the

anisotropy is small, the square root in equation (2.1) may be expanded

[20] and the energy surfaces may be expressed by


ii2k2
E = Ev k (A B')j(e,f) (2.2)
v 2m
0


where












0 and ( are the spherical coordinates, EV is the top of the valence

band, and


j(I,4) = 1 + I Y[sin46(cos 4 + sin4 ) + cos 4 2/3] (2.4)


with


Y = C C2/2B'(A + B') (2.5)


Following the work of Barber [25], we have used the simplified

model of the band structure illustrated in Figure 2.1. In this model

the heavy-hole band is considered parabolic and thus the mass mT is a

constant, equal to its value at 4.2 K. For energies within 0.02 eV

the light-hole band is considered parabolic with a constant slope

corresponding to the value of m* at 4.2 K. For higher energies the

light-hole band is assumed to take on approximately the same slope as

that of the heavy-hole band, but remains separated from the heavy-hole

band by A/3 eV [27]. The extrapolation of these two constant slopes

creates the kink in the light-hole band at 0.02 eV. Because of the

change in slope, the light-hole band has an energy-varying effective

mass and in general can only be described in terms of partial Fermi-

Dirac integrals [25]. Although the split-off band is parabolically

distributed, the apparent effective mass at the top of the valence band

is a function of temperature due to the energy displacement at k = 0.

Theoretical and experimental studies [33,34] have shown that at high

temperatures the heavy-hole band is not parabolic and thus m* is not

energy and temperature independent. However, within the range of











0.00





0.02


0.04


0.06





0.08




















Figure 2.1.


K2 (Arbitrary)


Simplified valence band structure of silicon based on
Kane's [27] calculations and measured properties of the
valence band.








temperatures considered here, the assumption of parabolicity for the

heavy-hole band based on Kane's model [27] is reasonable. Other studies

[35,36] support the validity of-this model for the valence band of

silicon.

2.3 Effective Mass Formulation

In the case of spherically symmetric energy surfaces all of the

carriers respond in the same way to a given set of applied.forces. The

effective mass then acts as a scalar and thus has the same value for

the Hall effect, conductivity, and density of states. For nonspherical

energy surfaces, however, this is not the case. The mixed response of

carriers to a set of applied forces is reflected in differences between

the different kinds of effective masses. The density-of-states effective

mass, m*i, is defined from the relationship


p 2k Tm* 3/2
pi ih2i F1/2(n) (2.6)


where

i.
2() + exp(c-n) (2.7)


E = (EV E)/k T, n = (EV EF)/k T, k is the Boltzmann constant, EV is

the top of the valence band, and i = 1, 2, 3 refers to the heavy-hole,

light-hole, and split-off bands, respectively.

The electric current density in the presence of electric and magnetic

fields can be expressed by [20]


Jj = ajkEk + CjkkEkH + Ojk.mEk H Hm + ...


(2.8)








where Ek, H Hm are the electrical and magnetic field components and

the o's represent single-energy-surface conductivity coefficients. The
first coefficient in equation (2.8) is the zero-magnetic field electri-

cal conductivity, and the second coefficient is associated with the

nondirectional Hall effect. In the limit of weak fields the expansion

can be limited to the first two terms. We use the electrical conduc-

tivity coefficient, OC, to define the conductivity effective mass m*i,

by the relationship

e2
Ci = lli= =Pi mi (2.9)


and the Hall effect coefficient, OH' to define the Hall mobility effec-

tive mass by means of [37]


e3<2
OHi = 123i = Pi (m-i)2 (2.10)


To solve for m^i, mMi, and m*i, equations (2.6), (2.9) and (2.10) are
equated to the following expressions for pi, ojk' and ajkZ:


Pi 1 fo(k)d3k (2.11)
473

2 af
S e o 3E E 32.12)
jk 4T32 a E kAj kk


3 af
k e = fo DE E 3E d3k (2.13)
jk 43i4 fTp q kk (2.3)








where f is the Fermi-Dirac distribution function and c pq is the permu-
tation tensor. Since equations (2.11) through (2.13) do not assume an
effective mass, they are valid both for parabolic and nonparabolic band
structures. These equations are then evaluated for the model described

in Section 2.2.

This procedure yields single m*i, .mi, and mi for an equivalent

model which is isotropic and parabolic. These values, in general, will
be temperature and carrier-concentration dependent. Although equations

(2.6) and (2.11) through (2.13) are expressed in terms of Fermi-Dirac

statistics to stress their generality, conductivity and Hall effective

masses were derived using Boltzmann statistics to simplify the form of
the equation. To obtain values of m*i and mi we also require a proce-
2
dure for evaluating and in equations (2.9) and (2.10). This
will be discussed in Chapter III. The following sections present the
expressions for the effective masses in the individual bands.

2.3.1 The Heavy-Hole Band

In this band, the effective masses are given by


Mm* [f(-Y)]2/3 (2.14)
mD1 (A-B')


mI A f(-Y) (2.15)
ml TA-(-P fl(- y

and

me =(A-B f2 (2.16)


where y is defined in equation (2.5). In these equations


f(y) = (1 + 0.05y + 0.01635y2 + 0.000908y3 + ...)








fl(y) = (1 + 0.01667y + 0.041369y2 + 0.00090679y3


+ 0.000919594 + ...)

and


f2(y) = (1 0.01667Y + 0.017956y2 0.0069857y3


+ 0.0012610y4 + ...)

Since the heavy-hole band was assumed parabolic, the integrals containing
T in equations (2.9) and (2.10) are identical to those in equations (2.12)
and (2,13), and cancel out.

2.3.2 The Light-Hole Band
In the light-hole band, as modeled by Figure 2.1, the effective
masses of holes are obtained in terms of partial Fermi-Dirac integrals
[25]. Thus

2m 3/2 f/koT 0 .
(m* )3/2 o f(+Y) 1 E'de
2 (A+B')3/2 o exp()


f(-Y)n o ed 1
S/ exp(EJ (2.17)
(A-BI) 3/2 /k T exp(c)








S2m T2 3/2de
C2 f o exp(c) x

f(+Y) /k T 4d- f -Y 1 m E dE1
(A+B')3/2 o exp() (A-B')3/2 '/kT p-


f (+Y) //koT T23/2de f (-Y)n r o To 3/21de
(A+B1 1/2 -f 0 xp E) + I ep1 1
(A+B') o exp() (A-B')1/2 /k T exp(e)


(2.18)



S(A+B')1/2 f /k T 223/2exp(-E)dE +


(A-B')12 2(-Y)1 f T 22 3/2exp(-1)d
&/k0 T
m* = m
H2o 3/2 /k f(+Y) f /koT
F o exp() (A+B')3/2 exp(c)

f(-Y)Tnl co el de
(A-B')3/ 2 /k2 T f xp J (2.19)
o/koT

where el = A/3koT, C = A A/3, ql = exp(-A/3k T) and A and 5 are
defined in Figure 2.1.
In this case because equations (2.11) through (2.13) were
expressed in terms of partial Fermi-Dirac integrals and equations (2.6),
(2.9') and (2.10) were expressed in terms of complete Fermi-Dirac
integrals, the dependence on T does not cancel out. Thus the nonpara-
bolicity of the light-hole band introduces a dependence on the scatter-
ing relaxation time. The scattering relaxation time is discussed in
Chapter III.








2.3.3 The Split-Off Band

Although the split-off band is parabolic, the apparent effective

mass in this band will also exhibit a temperature dependence due to the
energy displacement at k = 0. The energy of a hole in the third band

is given by

2k2
E = E A A (2.20)
v 2m0


where A is the split-off energy (= 0.044eV), and A is one of the inverse
mass band parameters. Substituting equation (2.20) into equations
(2.11) through (2.13), and then equating to equations (2.6), (2.9) and

(2.10) for the split-off band, we obtain

m m 2A (2
AD3 exp (- _3)T (2.21)



f T 33/2exp(-c)de
m* 0- m (2.22)
3/2
j T3 2 exp(-e2)d2
0



m J T3 e 3exp(-c)de
m*3 = A 0 (2.23)
/ T3 23/2exp(-E2)d c2



where E2,.= A/k T.
The combined hole density-of-state effective mass can be determined

by assuming that the total number of holes in the valence band is equal
to the sum of the holes in the individual bands








P = pl + P2 + P3 (2.24)


thus


m = [(m +)3/ (2 + (m3 3/2]2/3 (2.25)


This combined effective mass is the mass corresponding to the density-

of-states of an effective single equivalent parabolic valence band.

This concept is useful in calculations where the effective density-of-

states at different temperatures can be calculated from one m.I

The explicit temperature variation of the band curvature is included

by assuming that the density-of-states near the band edges varies in a

similar manner as the temperature dependence of the energy gap [25].

Thus (m*)3/2 is porportional to EGo/EG where EGo is the energy gap at 0 K.

To evaluate the total band equivalent conductivity and Hall effec-

tive masses, we assume that in valence band conduction, the total number

of holes in motion is equal to the sum of the holes moving on the

separate energy surfaces, and that these holes can be modeled as moving

on a single spherical energy surface. Thus, the ohmic and the Hall

conductivities in the equivalent valence band are given by

aC = Cl1 + C2+ C3 (2.26)


and


H = H + H2 + aH3 (2.27)


respectively.








Substituting equations (2.9) and (2.10) into equations (2.26) and

(2.27) it follows that


S imf 3/2 I <2> m2 3/2 <3> m3 3/2 1 -1
C m1 mC2 mD J mC3
(2.28)

and

S2 3/2 m* 3/2 < 2> 3/2 1
Hm {<2> .2 2 mm
m< [2> .2n m 2 2. i m>2 <2 > 3J 2
D H H2 H3
(2.29)


Equations (2.25), (2.28) and (2.29) were evaluated numerically as func-

tions of temperature and acceptor doping density for p-type silicon.

Values of the band parameters, IAI = 4.27, IBI = 0.63 and ICI = 4.93,

were determined at 4.2 K by Hensel and Feher [22] and Balslev and

Lewaetz [29]. In order to simplify the calculations and maintain

tractability, anisotropies in the relaxation time were ignored. A rig-

orous analysis of the conductivities for nonisotropic scattering would
be extremely difficult to carry out because no relaxation time is

expected to exist in the usual sense [38].
Figure 2.2 shows the dependence of m* with temperature in the range

from 100 to 400 K. The slight temperature dependence due to the expli-

cit temperature variation of the.curvature at the edge of the band

results in an effective mass increase of about 5 percent in each

band at 400 K. This can be seen in the slope of m1l. The temperature

dependence of m*3 is more pronounced since here we also have the

effects of energy displacement at ~ = 0. The temperature dependence due

to nonparabolicity is very apparent in the shape of the mD2 curve.













0.90



0.80



0.70


0.60



0.50



0.40



0.30



0.20


0.10



0.00
1


Figure 2.2.


100


150 200 250 300 350 400
Temperature (K)


Temperature
masses m*1,
Dl'


dependence of the density-of-state effective
m*2 and m*3, in the individual bands, and the
D'2 "D3'


combined density-of-states mass m* of holes in silicon.
14 -3
N = 10 cm .
A








The temperature dependence of the conductivity effective mass and

the Hall effective mass is shown in Figures 2.3 and 2.4, with the dopant
14 -3
density equal to 10 cm One consequence of the nonparabolicity of

the light-hole band is an increase in the valence band conductivity

effective mass as temperature increases from 100 to 400 K. This happens

because with increasing thermal energy k T, more holes reside in the less

parabolic regions of the light-hole band. The results plotted in Figure

2.3 show an increase in m* of about 36 percent in this temperature range.

The temperature dependence of mH can be attributed mainly to the non-

parabolicity of the light-hole band. In the temperature range from 100

to 400 K, mj increases from 0.2850 to 0.5273 m The slight temperature

dependence of m*l and m"H is due to the explicit temperature effect and

results in increases of 7.7 percent and 3.76 percent in the m*l and mHl

respectively. A larger temperature variation occurs in the case of the

split-off band because of the additional effects of the energy displace-

ment at k = 0.

Figures 2.5 and 2.6 show the variation of m* and mH with dopant

density and temperature. For T 2 100 K, mC varies less than 10 percent
14 18 -3
in the dopant density range from 101 to 101 cm3. Since the influence

of nonparabolicity is reduced in degenerate material [25], it follows

as shown in Figures 2.5 and 2.6 that the variation of effective mass

with temperature is much stronger at low dopant densities. At lower

temperatures there is a much greater change in effective mass due to

variations in scattering relaxation time with percentage of ionized

impurities.























0.60


E 0.50




0.40



4-)
0.30




0 0.20 "
4-)









0.10
100





Figure 2.3.


Temperature (K)


Temperature dependence of the conductivity effective masses
m*, mC2 and mC3 in'the individual bands, and the combined
conductivity effective mass m* of holes in silicon.
14 -3
N. = 10 cm .







23


100 150 200 250 300 350


Temperature (K)


Figure 2.4.


Temperature dependence of the Hall effective masses, m*,

m*2 and m*3 in the individual bands, and the combined
S14 -3
Hall effective mass m* of holes in silicon. N = 10 cm .
H A


0.60


0.50


0


ro
E



59c,


E


>
*r~
4-,
U
0)
4-
4+-
u-I













0.50


0.40


0.30


400 K


350 K


300 K
I


200 K.


250 K







150 K




100 K


1015


Figure 2.5.


1016

NA(cm3)


1017


1018


The acceptor density dependence of the combined conductivity effective mass
of holes in silicon as a function of temperature.


0


Ip


1 - -


-- -






















o V".-- 150 K




cn
0.30. 100 K





0.20



1014 1015 1016 1017 1018

NA(cm-3)


Figure 2.6. -The acceptor density dependence of the combined Hall effective mass of holes in
silicon as a function of temperature.








2.4 Discussion

The idea of temperature-dependent effective mass is supported by

a number of experimental data. Cardona et al. [21] found an increase of

about 12 percent in optical effective mass between 90 and 300 K in

heavily-doped p-type silicon. Cyclotron-resonance studies conducted by

Hensel and Feher [22] show that when carrier heating populates deeper

regions'of the light-hole band, the nonparabolic nature of this band at

higher values of T results in an increase in the effective mass of holes.

The model used here in the calculation of hole density-of-states-

effective mass is identical to that of Barber [25], and consequently

our results for m*i and m are in excellent agreement with those of

Barber [25]. We have extended Barber's work to the calculations of mC

and mH in p-type silicon. The increase of m* by 36 percent at 400 K

shown in Figure 2.3 is much larger than that reported by Costato and

Reggiani (9 percent) [26]. Their calculation was done over a similar

range of temperatures, and their value at 100 K, mC=0.342 m is some-

what lower than our calculated value (.3604 m ). The discrepancies

between our results and those of Costato and Reggiani are due mainly to

the correction of mD for the explicit temperature dependence of the

energy gap, the inclusion of the split-off band, and the consideration

of unequal relaxation.times in the three bands. Note that our calcula-

tions of effective masses were achieved through more rigorous mathe-

matical derivations, while those of Costato and Reggiani followed a

more empirical curve-fitting type of procedure.

The experimental values of density-of-states effective masses of

holes in p-type silicon have been published by numerous authors [21,22,

39,40], but very little data can be found for the conductivity and the








Hall effective masses, making it difficult to properly assess the value

,of our calculations. There seems to be no obvious way to measure these

quantities from d.c. transport measurements. Magneto-kerr effect

measurements conducted by Hauge [41], indicate that mC could increase

by as much as 31 percent in the range of temperatures from 100 (M* =

0.510 Mo) to 300 K. This is i.n reasonable agreement with our calculated

percentage increase in m* in the same temperature range (33 percent),

but it is impossible to compare our calculations with Hauge's experi-

mental results, because our effective mass definition was chosen to be

mainly applicable to the study of the Hall and conductivity mobility in

the low field limit, and this may not apply to the measurements of

Hauge [41].

From'the results of this chapter it can be seen that the approxima-

tion of a constant effective mass seems to be inadequate to describe

transport properties of holes in silicon above 100 K. There is a sub-

stantial increase in the effective mass of holes from 100 to 400 K due

to the nonparabolicity of the light-hole band, and a smaller, though not

negligible, contribution due to the explicit temperature dependence and

the effects of the split-off band. The validity of this model for the

calculation of density-of-states effective mass has been well established

[25]. Barber [25] has shown that when the temperature-dependent effec-

tive masses are substituted into the theoretical expression for intrinsic

carrier density in silicon, the agreement with reported measurements of

ni is within the limits of experimental error. Application of this

model to theoretical calculation of mobility and resistivity in p-type

silicon [17] has provided excellent agreement between theoretical and

and experimental values (resistivity with 6 percent) over a temperature




28


range from 100 to 400 K and dopant density range from 1014 to

3x1018 cm-3. This calculation is limited to applications in conduc-

tivity mobility and low field Hall effect.













CHAPTER III

MOBILITY AND SCATTERING RELAXATION TIME

3.1 Introduction

The study of transport phenomena in semiconductors requires an

accurate knowledge of variations in the conductivity mobility and the

resistivity with changes in temperature and dopant density. The resis-

tivity is an easily-measured parameter, but the conductivity mobility

is a more difficult parameter to evaluate. In general, four different

kinds of mobility enter into common discussion [42]. The microscopic

mobility is the actual velocity per unit electric field of a free

carrier in a crystal. This cannot be measured directly. The conduc-

tivity mobility is the mobility associated with the conductivity

expression, a = eppC. This mobility involves an average relaxation

time dependent on the nature of the scattering process, and in the

case of nonspherical equal energy surfaces, this mobility also involves

a combined effective mass. The Hall mobility is the product of the

measured conductivity and the measured Hall coefficient. In general,

the Hall mobility differs from the conductivity mobility by a factor

called the Hall factor. The drift mobility is the velocity or drift

per unit field for'a carrier moving in an electric field. If trapping

centers are present, so that the actual drift process is not simply

motion through the conduction band, but involves a series of trapping

and untrapping processes, the drift mobility can be much less than the

conductivity mobility. The four mobilities are all equal only when the

29








following three conditions are met [42]: (a) spherical equal energy

surfaces with extremum at 0 = 0, (b) relaxation time independent of

carrier energy, and (c) negligible trapping effects. Since conditions

(a) and (b) are not met in'p-type silicon, it is improper to judge the

behavior of one kind of mobility based on knowledge of a different

kind of mobility. Thus drift or Hall mobility data cannot be tacitly

assumed to be accurate substitutes for conductivity mobility values.

As mentioned above, the conductivity mobility involves an average

scattering relaxation time. In any semiconductor, the charge carriers

(i.e., holes and electrons), at temperatures above absolute zero, may

be scattered by a number of mechanisms. Different mechanisms are

dominant in certain temperature and dopant density regimes, but in some

cases two or more may be interacting simultaneously. Thus in calculat-

ing the conductivity mobility over a wide range of temperatures and

dopant densities, all the relevant scattering mechanisms must be taken

into account. In the case of silicon, acoustic and optical phonon

scattering, and ionized and neutral impurity scattering are of major

importance. Hole-hole scattering also plays an important role in deter-

mining the mobility. In the following sections the theoretical effects

of these scattering mechanisms on the mobility will be considered.


3.2 Mobility and Average Scattering Relaxation Time

The calculation of mobility of holes in the valence band of silicon

is accomplished by evaluating the mobility separately in the heavy-hole

band, the light-hole band, and the split-off band considering all

appropriate scattering mechanisms. The overall mobility is then

evaluated as a weighted average of the single-band mobilities over the

individual hole densities in each band.








The conductivity mobility in each of the three valence bands is
calculated from

e
Ci m (3.1)


where

r 3/2 fo dl
C T "')j
= d (3.2)
S3/2 fo
jE; 9 dc


for the case of Fermi-Dirac statistics, and Ti represents the total
scattering relaxation time in band i. Because each scattering mechanism
has its own dependence on scattering energy, a simple closed form
expression for total scattering relaxation time as a function of temper-
ature cannot be obtained. The use of numerical techniques is necessary
to solve for the relaxation time. In the case of p-type silicon, the
peculiarities of a degenerate, warped, and nonparabolic valence band
must.be taken into account [1]. The possibility of interband as well as
intraband transitions must also be taken into account in the analysis.
With the inclusion of interband scattering as given by Bir et al. [43],
the total relaxation time in the heavy- (i = 1) and light-holes (i = 2)

bands is given by


T I + -~i -rI, i j; i = 1,2; j = 1,2 (3.3)
D lJ ,


where








11 22
6 = 1 -(3.4)
'12 21

and


T -1 -1 + -1 + 1 T1 (3.5)
Sii = [aci ?oi +li Ni


The total relaxation time in the split-off band is given by


o3 = [+ 3 + T13 + TN3 -1 (3.6)


Only transitions between the light- and heavy-hole band are con-

sidered; the relaxation time T.. takes into account a transition from

band i to band j; and Taci Toi, T Ii and TNi are the relaxation times

corresponding to scattering by acoustical phonons, optical phonons,

ionized impurities, and neutral impurities respectively, with i as the

band index. The procedure for including the nonparabolicity of the

band structure into calculations of relaxation time consists of

modifying the relaxation time for a given scattering process by replac-

ing the temperature independent effective mass of the parabolic band by

the temperature dependent effective mass of the nonparabolic band.

This procedure has been successfully applied to the study of acoustic

phonon scattering in nonparabolic bands by Radcliffe [18]. Optical

phonon and ionized impurity scattering in nonparabolic bands have been

considered by Barrie [19] in the same manner. Braggins [1] has used

the same method to include nonparabolicity in his study of p-type sili-

con. In this work, the relaxation times appropriate to degenerate,

parabolic valence bands have been used and modified according to the

prescription of Radcliffe [18], Barrie [19], and Braggins [1]. The








anisotropy of the energy spectrum is not considered in this model,
because from the transport theory for parabolic bands it is known that

this anisotropy has no influence on the temperature dependence of

mobility, but only on its absolute value [10]. Each of the four
scattering mechanisms will now be discussed.

3.3 Acoustical Phonon-Scattering

The relaxation time for scattering by acoustical phonons includes
both the possibility of interband as well as intraband scattering. The

treatment of the acoustical phonons has been based on the theory of

Bir, Normantas, and Pikus [43] where the relaxation times can be
expressed in terms of a single constant, Tx, which controls the overall
magnitude of the scattering. Both transverse and longitudinal phonons

participate in the scattering so that

3/2
-1 m-1 L (2) + 3 (1)
acl T-x 11 1ij 11



1 +2 T T 1i) T} (3.7)
Ct


and

3/2
-1 D2 (2) 3 L22(1)




C 2 [T22(2)+ 3 (1)) T3/2cI/2 (3.8)
t J 2








for intraband scattering, while

S3/2
-1 -5 -1 i L (2)+
ij Y.. Ti .. L.
x2

S82 T(2)] } T3/2 1/2 (3.9)

t


for. interband scattering. In the split-off band, the scattering relaxa-
tion time is given by

-1. 1 1/2 3/2
ac3 T T (3.10)


In these equations


k 3/2a2 m3/2
1 o o (3.11)
x '1Ti4p C2
s 2

Ytj = mDi/mj, = b/a, a and b are valence band acoustic deformation
potential constants in the Picus and Bir [44] notation, p is the
density, CP and Ct are the longitudinal and transverse sound velocities

in silicon and Lij and Tij are functions of B and Yij defined in [43].

3.4 Optical Phonon Scattering

Optical phonon scattering, while negligible at very low tempera-
tures, cannot be ignored at high temperatures. Ehrenreich and

Overhauser [45] have calculated the mobility of holes in silicon and its
dependence on temperature. The calculated mobility follows a T-2.3

dependence for reasonable choices of the parameters which described the








mixing of optical and acoustical phonon scattering. This agrees with
experimental results [5,8]. The relaxation time for scattering by

nonpolar optical phonons is given by [46]


3/2 a _112
-1 m*i 1/2 D (n+l)r 1/2
Toi DT (n +1) C/ +
oi x D


r Y1/2 (
n + } i = 1,2,3 (3.12)


where 0D is the Debye temperature, n = (exp(6D/T)-1)- is the phonon

distribution function, and W is a constant which determines the rela-

tive coupling strength of the holes to the optical phonon mode compared

to the acoustical phonon mode


D 2 2C 2
W o (3.13)
2k a 02

2
'where D is the optical deformation potential constant. The first term

in the brackets of equation (3.12) corresponds to optical phonon emmis-

sion and is relevant only when this is energetically possible (e>eD/T).

The second term in the brackets corresponds to optical phonon absorption.

3.5 Ionized Impurity Scattering

The Columbic interaction between ionized impurities and charge

carriers drifting through the cyrstal under the action of an applied

electric field causes scattering of the charge carriers. Scattering by

ionized impurities was first considered by Conwell and Weisskopf [47].

The basic assumption is that the Coulomb field is cut off at half the








distance between charged impurities. This is equivalent to assuming

that a charge carrier sees only one charged impurity at a time, the

effect of the other chargedimpurities being sufficiently screened as

to be negligible. This approach was improved by Brooks [48] and

Herring [49] who associated the cut-off of the Coulomb potential with

a screening distance, the free carriers being assumed to provide

screening against the charge of the impurities. In the low dopant

density limit, the scattering relaxation time due to ionized impurities

is given by [48,49]


re NAG(b) -3/2
li (2m /2 s2(kT)3/2 /2 i = 1,2,3 (3.14)
Ii (2m)/2,(kT)3/2


where


G(bi) = n(bi+l) (b- +) (3.15)


and-

24T m*i.s(k T)2
b. 2 s (3.16)
S e2h2 p


where p' is the screening carrier density, p' = p + NA(1 NA/NA), for

ND = 0.

3.6 Neutral Impurity Scattering

Scattering by neutral impurities in semiconductors has been con-

sidered by Erginsoy [50] as a variation of the problem of the scattering

of electrons by neutral hydogen atoms. The result is a temperature

independent relaxation time given by








2-
-1 C 2 0i'
i mi NN i 1,2,3 (3.17)
mGe Di


where NN is the density of'neutral impurities and mG is the geometric

mean mass appropriate for evaluating the scaled Bohr radius term [48].

Sclar [51,52] has included the possibility of bound states in the

evaluation of electron-hydrogen impurity scattering by using a three-

dimensional square well to estimate the influence of a weakly-bound

state on the scattering. In this case the relaxation time is given by


23/2 T 2N E
Ni 1/2 3/2 + T1/2 i = 1,2;3 (3.18)
(k 0T) mDi k T

where 2

E 1.136 x 10-19 D ) (3.19)


is the binding energy of neutral acceptors.

For silicon doped with shallow impurities, this type of scattering

is important at low temperatures where neutral impurities may outnumber

ionized impurities. For the deeper levels, where neutral impurities can

exist at higher temperatures, the influence of neutral impurity scatter-

ing can extend over a wide range of temperatures.


3.7 Effect of Hole-Hole Scattering

The expressions thus far presented for scattering relaxation time

neglect the effect of hole-hole scattering. Although hole-hole scatter-

ing does not affect the current density directly since it cannot alter

the total momentum, it tends to randomnize the way in which this total







momentum is distributed among holes of different energies. When the
scattering mechanism is such as to lead to a nonuniform distribution,
hole-hole scattering gives rise to a net transfer of momentum from
holes which dissipate momentum less efficiently to those which dissipate
momentum more efficiently, resulting in an overall greater rate of
momentum transfer, and lower mobility [53]. Thus the size of the effect
of hole-hole scattering on the scattering relaxation time is a function
of the energy dependence of the relaxation time. The hole-hole reduc-
tion factor, Yhh, can be derived by means of a classical formulation
introduced by Keyes [54]. When hole-hole collisions are much more
frequent than hole-acceptor collisions, the average relaxation time for
a parabolic band in the Keyes [54] approximation approaches the
limiting form


S3/2 foL
<'hh> = (3.20)
f 3/2 -1T o 0
f T i

where f is the Fermi-Dirac distribution function. On the other hand,
if hole-hole collisions are neglected, the average relaxation time is
given by equation (3.2).
Thus the hole-hole reduction factor (i.e., the ratio of to
) can be expressed as

(f f -I
S3/2 F 3/2 -1L
f e T 'oJ dc x fe C T dc
Yhh P2 -fd (3.21)
/ ar-j dI]
I U ^~r .








Yhh 1 (3.21a)
Yhh -


for optical phonon scattering, and Yhho, the hole-hole reduction factor,
is evaluated 'from equation (3.21).
For acoustical phonon scattering it is assumed that Yhha decreases
linearly with increasing dopant density from a value of one to a value

hha = 97/32 = 0.88 [17] in a certain range of impurity concentration.
(a -19 15
The exact relationship (yhh = 1.0004 4.013378 x 10 NA, 1015 NA
3 5 1017) is determined empirically with a best fit of the experimental
data.
Luong and Shaw [55] using a one-particle-like approximation from
the Hartree-Fock theory, have shown that by inclusion of hole-hole
scattering, the Brooks-Herring [48,49] formula is reduced by a factor
which can be expressed in closed form as


Yhh ; -I exp2] (3.22)
N

where NA-is the ionized acceptor density and p' is the screening hole
density. In the case of neutral impurity scattering, hole-hole scat-
tering has no significance because TN is independent of hole energy.

Thus the overall scattering relaxation time in each hole band is
calculated from equations (3.3), (3.5), and (3.6) with the terms of
these equations properly corrected for the effects of hole-hole scatter-
ing. Because the individual energy surfaces are different from each
other, the relaxation times also differ from each other and cannot be
assumed equal except in restricted ranges of temperature and dopant
density [43].





40


3.8 Mobility in the Combined Valence Band

The conductivity mobility in each individual band is calculated

from equation (3.1), and the combined conductivity mobility in the

valance band is then evaluated as a weighted average of the single-band

mobilities over the population of holes in each band, thus


mnC l 3/2 m 2 3/2 m3 3/2
: lj + D2 ) + 3mJ D(3.23)



Using equation (3.23) and the parameters listed in Table 3-1, we

have calculated the hole mobility for silicon doped with boron, gallium,

and indium as functions of dopant density and temperature, for

1014 NA 10 3cm-3 and 100 < T < 400 K. The results are displayed in

Figures 3.1 through 3.6. In the calculations of mobility and resistivity

in silicon doped with gallium and indium, it was assumed that boron

impurities were also present. Since very pure silicon has a resistivity

on the order of 1000Q-cm, it was assumed that boron densities of 1013
13 -3
and 5x10 cm existed in the gallium- and indium-doped samples,

respectively. The values of these background densities were deduced

from a best fit of the experimental data. For this reason, especially

in the case of indium-doped silicon, the actual role of the impurities

at low temperatures and/or low dopant densities is masked by the action

of the always present boron impurities. As the dopant density and

temperature increase, the assumed background densities of boron

impurities become insignificant compared to the density of ionized dopant

atoms, and Figures 3.1 through 3.6 accurately depict the influence of

the particular type of impurity on the resistivity and mobility of holes

in p-type silicon. The figures also show that for the case of the










Table 3-1. Values used in the calculations.


* These values were obtained


from references [1] and [22].


--


---


Parameter

A

a

b

C 2/Ct2

0D

PS

Es

Tx

W

m

h

k

e


Value

44.0

-6.4*

-1.36*

2.09*

735

2.329 x 103

11.7

6.96 x 10-10

0.244

9.1 x 10-31

6.25 x 10-34

1.38 x 1023

1.6 x 10-19


Unit

meV

eV

eV



K

kg/m3

0
sec K3/2



kg

joule-sec

joules/K

coul











































Boron Density (cm-3)
Boron Density (cm )


Figure 3.1.


The calculated hole mobility vs dopant density for boron-doped silicon with
temperature as a parameter.


0


U



U


-Q


a)
0


102


















































1018


Gallium Density (cm-3)


Figure 3.2.


The calculated hole mobility vs dopant density for gallium-doped silicon with
temperature as a parameter.


0


u
C)
>
vC,

(N
E


+-3
,I-

0
0)
Cr
0
3I


101





0


1015 1016 1017 1018
Indium Density (cm-3)


Figure 3.3.


The calculated hole mobility vs dopant density for indium-doped silicon with
temperature as a parameter.













































150 200 250 300 350


400


Temperature (K)


Figure 3.4.


The calculated hole mobility vs temperature for boron-doped
silicon with dopant density as a parameter.


100












































150 200 250 300 350 400


Temperature (K)


Figure 3.5.


The calculated hole mobility vs temperature for gallium-
doped silicon with dopant density as a parameter.









































150 200 250 1 300 350


400


Temperature (K)


Figure 3.6.


The calculated hole mobility vs temperature for indium-
doped'silicon with dopant density as a parameter.


104


100








shallower ionization energies, the mobility depends more strongly on

temperature for the lightly-doped case where lattice scattering is

dominant and become less temperature dependent as the dopant density

increases.

The constant, Tx, was found by fitting the mobility to experimental

data in the lattice-scattering-limited range. Our value of Tx is

equivalent to an acoustic deformation potential constant of 8.099 eV.

The optical phonon coupling constant, W, was then found by fitting the

mobility to the high temperature experimental data. Our value of W is

equivalent to an optical deformation potential constant of 6.024x108 eV/

cm.













CHAPTER IV

HOLE DENSITY AND RESISTIVITY

4.1 Introduction

The resistivity of semiconductor materials is one of their most

useful and easily measured properties. Theoretical calculations of

resistivity depend on the formulation of conductivity mobility, and the

determination of hole density. For extrinsic semiconductors, the hole

density is determined primarily by the percentage of ionization of

impurity atoms. The following sections discuss the dependence of hole

density and resistivity on temperature and dopant density.

4.2 Ionization of.Impurity Atoms

For the case of Fermi-Dirac statistics, the hole density is given

by

4 2k Tm* 3/2
p h2 D] F(Wn) (4.1)
n' h2 '2


where, m*, the density-of-states effective mass, contains information

pertaining to the nonparabolic nature of the valence band. In the limit

of low dopant densities, equation (4.1) reduces to


p = NVexp(n) (4.2)


where NV = 2(2 T m* k T/h2)3/2 is the effective density of valence, band

states. For the range of temperatures considered in this study, the

49








hole density i.s calculated by assuming that the density of carriers is

determined by the impurities present in the silicon sample. The density

of ionized acceptor impurities in p-type silicon is computed from the

charge neutrality equation

+
NA ND = p n (4.3)


This reduces to


p NA (4.4)


for the case of uncompensated material.

The density of ionized acceptors is [56]

NA
NA =__ (4.5)

1 + g exp [E T


where EA is the acceptor ionization energy, and g is the ground state

degeneracy. Excited states have a very minor influence on the carrier

concentration due to the large separation between the ground state and

the excited states [1,56]. Letting


g = 4 + 2 exp[- (4.6)


enables us to include the contribution of the split-off band [17]. The

density of ionized acceptors is computed by iterating EF in equations

(4.2) and (4.5) until equation (4.4) is fulfilled within a given level

of accuracy.








Experimental evidence shows that the acceptor ionization energy EA

is not a constant, but decreases with increasing dopant density [9].

Penin et al. [57] have determined in a study of heavily doped silicon

from 4 to 300 K that for shallow impurities such as boron and phosphorus

the ionization energy decreases and finally disappears altogether for

impurity densities greater than 3x108 cm3. For impurities with deeper

activation energies, it is also expected that at some impurity concen-

tration, the impurity activation energy should become a function of the

impurity concentration. However, in the case ofgallium and indium,

this should happen at higher impurity concentrations than for the

shallower level impurities. This is due to the smaller geometrical

dimensions of the wave functions applicable to the deeper levels, so

that overlapping effects which promote the reduction in activation energy

require higher impurity concentrations [14]. For shallow impurities

such as boron and phosphorus, empirical expressions [9,57] relating the

dependence of ionization energy to dopant density have been established.

In the case of Ga, there is data [15] on' activation energy vs concentra-

tion, but not enough on which to base an accurate relationship. For

this reason the value of EA = 0.056eV was used. For In, EA = 0.156eV

[58] was used. Figures 4.1 through 4.3 show the ratio of ionized and

total impurity density as a function of impurity density with tempera-

ture as a parameter for 100 s T s 400 K for silicon doped with boron,

gallium, and indium. It is clearly shown in these figures that the

ionization of impurities for the deeper levels is significantly lower

even at low dopant densities so that it is necessary to go to higher

































20



0
1014


1016
Total Boron Density (cm-3)


Figure 4.1.


Theoretical calculations of the ratio of ionized and total boron density
vs boron density with temperature as a parameter.


1018


















































1018


1015 1016 1017

Total Gallium Density (cm-3)


Figure 4.2.


Theoretical calculations of the ratio of ionized and total gallium density
vs gallium density with temperature as a parameter.


(2,
-o
N


4-


C-)
a,-


1014





0


Total Indium Density (cm-3)


Figure 4.3.


Theoretical calculations of the ratio of ionized and total indium density
vs indium density with temperature as a parameter.


1o00


1014


1015


1017


1018








temperatures to achieve total ionization of impurities. The deioniza-

tion of impurities is most significant for low temperatures and high

impurity densities.


4.3 Resistivity of p-Type Silicon

The resistivity of p-type silicon is given by


p (4.7)


where vC is the hole conductivity mobility calculated from equation

(3.23) and p is the hole density discussed in Section 4.2. Equation

(4.7) was used to calculate the hole resistivity for silicon doped with

boron, gallium, and indium as a function of dopant density and tempera-

ture, for 1014 < NA 1018 cm3 and 100 T 400 K. The results are

displayed in Figures 4.4 through 4.9. In the calculations of resistiv-

ity in silicon doped with gallium and indium, as was done for conductiv-

ity mobility, it was assumed that boron impurities were also present.

Boron densities of 1013 and 5x103 cm-3 were assumed to exist in the

gallium- and indium-doped samples, respectively. The values of these

background densities were deduced from a best fit of the experimental

data. As the dopant density and temperature increase, the assumed

background densities of shallow impurities becomes insignificant

compared to the density of ionized dopant atoms, and Figures 4.4 through

4.9 accurately depict the influence of the particular type of impurity

on the resistivity of holes in p-type silicon. The figures also show

that for the case of the shallower ionization energies, resistivity

depends more strongly on temperature for the lightly doped case where

lattice scattering is dominant and become less temperature dependent

as the dopant density increases.
















































150 200 250 300 350


Temperature (K)


Figure 4.4.


Theoretical calculations of resistivity vs temperature for
boron-doped silicon with dopant density as a parameter.


101





100


10-1





10-2


100.










































10-
10





10-2
100








Figure 4.5.


Temperature (K)


Theoretical calculations of resistivity vs temperature for
gallium-doped silicon with dopant density as a parameter.





























101





100


10-1


100






Figure 4.6.


150 200 250 300 350


400


Theoretical calculations of resistivity vs temperature for
indium-doped silicon with dopant density as a parameter.






























100


10-1





10-2


1015 1016 1017

Boron Density (cm-3)


Figure 4.7.


Theoretical calculations of resistivity vs dopant density
for boron-doped silicon with temperature as a parameter.


1018













































1016 1017

Gallium Density (cm-3)


Figure 4.8.


Theoretical calculations of resistivity vs dopant density
for gallium-doped silicon with temperature as a parameter.


10-1


10-2
1014


1015


1018






















101





100


10-1


1018


1016 1017

Indium Density (cm-3)


Figure 4.9.


Theoretical calculations of resistivity vs dopant density
for indium-doped silicon with temperature as a parameter.













CHAPTER V

THE HALL FACTOR IN p-TYPE SILICON

5.1 Introduction

The most direct determination of the mobility is by the Haynes-

Schokley drift method, wherein the drift of charge carriers in a known

electric field is measured. However, the assumption, made when these

experiments were initiated, that the drift mobility of holes as minority

carriers in an n-type sample is the same as when they constitute the

majority carriers, is invalid in view of carrier-carrier scattering

[59]. Also the experiment can succeed only if the lifetime of the

minority carriers is larger than the transit time. For this reason,

usually Hall mobilities are measured instead. The Hall mobility is the

product of the measured conductivity and the measured Hall coefficient.

In general the Hall mobility differs from the conductivity mobility by

a factor called the Hall factor. Determination of the Hall factor may'

be avoided by making use of the high field limit. For sufficiently

high magnetic fields several simplifications occur in the magnetic field

dependence of the Hall coefficient. In the high-field limit (when the

product of mobility and magnetic induction becomes greater than 108 cm-2

gauss/volt-sec [60]) the Hall coefficient is simply related to the

carrier concentration by [61]


1)
RH pe (5.1)








and thus the conductivity mobility and the Hall mobility are equal.

Although the high field limit simplifies use of the Hall mobility con-

siderably, excessively high magnetic fields can cause problems due to

the quantization of the hole orbits in a magnetic field [1]. The

quantization of the particle motion in a magnetic field will create

Landau levels within the band. The Landau levels will modify the den-

sity of states in the valence band which could affect the interpreta-

tion of experimental,data [56]. Another high magnetic field effect of

importance is the "magnetic freeze out" which occurs with the stronger

localization of bound state wave-functions in a strong magnetic field

[62]. Due to the more localized charge distribution, the Coulomb

binding energy of the impurity state is increased so that at a fixed

temperature the concentration of thermally excited charge carriers will

be smaller and the Hall coefficient will be effectively increased.

Thus, in order to avoid these high field region complications and

obtain an experimental determination of the value of conductivity

mobility in the low field limit, it is necessary to have an accurate

knowledge of the Hall factor with which to modify measured Hall

mobilities. Hall measurements are routinely used to experimentally

determine the density of ionized impurities in a semiconductor sample.

This determination is possible only if an accurate value of the Hall

factor for the particular temperature and dopant density considered is

available.


5.2 The Hall Factor

The Hall and conductivity mobilities are related by the Hall factor

as follows:






64


rH (5.2)
H pC


For nondegenerate, spherically symmetric bands, it can be shown that

[59] rH 1, and that


rH = (5.3)


In general most previous work [5,59] has assumed that equation

(5.3) is valid in the case of p-type silicon and thus rH will vary

between 1.18 (T(E) E1/2 for lattice scattering) and 1.93 (T(E)

E3/2 for ionized impurity scattering), if hole-hole scattering is

neglected, and will approach unity for the degenerate case. This theory

does not allow for values of rH less than one. Experimental evidence

indicating values of rH less than one has been attributed to poor

quality of the measured samples [5]. Debye and Kohane [63] found that

the measured drift mobility for holes is considerably larger than the

measured Hall mobility. Values of rH less than unity were also reported

by Wolfstirn [15] for the case of gallium-doped silicon. More recent

experiments [64] show that a value of rH less than unity is necessary

to reconcile differences between the hole concentration measured via

Hall coefficient .methods and that inferred from dopant densities

determined from C-V and junction breakdown measurements. The usual

assumption made is to let rH beequal to one and thus consider the Hall

mobility equal to the conductivity mobility. Neglecting the Hall

scattering factor alters both the magnitude and temperature dependence

of the carrier concentration from that given by the charge balance

equation. In fitting data to the charge.balance equation, both thermal








carrier concentration and dopant impurity activation energy are over-

estimated by the assumption of unity Hall factor. A more complete

theoretical treatment of the Hall factor can be undertaken by consider-

ing the nonparabolic and anisotropic nature of the valence band of

silicon.

Chapter II described the constant energy surfaces as warped spheres.

Warping of the energy surfaces has a significant effect on the ratio of

Hall to conductivity mobility. When the bands are warped, the Hall

factor depends on the degree of warping as well as the scattering

mechanism [30].

The Hall mobility is the product of the ohmic conductivity and the

Hall coefficient


"H = CRH (5.4)


In the low field limit the Hall coefficient for a nonparabolic,

anisotropic band i is given by [37]


RHi Hi2 (5.5)
Ci


Thus by substituting equations (2.9) and (2.10) into equation.(5.5) the

Hall coefficient can be expressed as


Hi rHi (5.6)
Hi pie


where








2 2
rHi = ml HJ (5.7)



is the Hall factor. We see that allowing for a difference between the

values of conductivity and Hall effective masses due to the anisotropic,

nonparabolic nature of the band, enables us to separate the Hall factor

into two components: the mass anisotropy factor given by


2
rAi (5.8)


and the scattering factor given by



rSi 2 (5.9)



These components of the Hall factor will be considered in detail in the

next two sections.

5.3 The Mass Anisotropy Factor

Lax and Mavroides [20] have derived expressions for rA based on the

Dresselhaus et al. [28] model of the valence band of germanium and sili-

con. Their formulation for rA acknowledges the anisotropy, but neglects

the nonparabolicity of the bands. In general it is found that rA is less

than unity unless the scattering anisotropy becomes extreme [30]. In

order to determine the variation of the mass anisotropy factor with

changes in temperature and dopant density for the combined valence band

of silicon, equation (5.8) was evaluated using the values of combined

valence band effective mass obtained from equations (2.28) and (2.29).








The results of this calculation are presented in Figures 5.1 and 5.2.

These figures show the significant contribution of the mass anisotropy

factor to the Hall factor. Since the influence of nonparabolicity is

reduced in degenerate material [25], it follows as shown in Figures 5.1

and 5.2, that the variation of rA with temperature is much stronger at

low dopant densities, since it is in this dopant density range that the

variation of effective mass with temperature is the strongest. We note

that the mass anisotropy factor is less than unity for all temperatures

considered in this work once the dopant density increases past
15 -3
6x105 cm3. At 300 K, rA is less than unity even for dopant densities
14 -3
as low as 10 cm .


5.4 The Scattering Factor

The scattering factor, rS, depicted in Figures 5.3 and 5..4 as a

function of temperature and dopant density, does not follow the tradi-

tionally expected variation between 3T/8 = 1.18 and 315T/512 = 1.93 as

.the dominant scattering mechanism changes from lattice to ionized

impurity scattering. Putley [65] has noted that hole-hole scattering

can modify rS. He estimates that for ionized impurity scattering, rS

can be reduced from 3157/512 to a value close to unity. At low dopant

densities where the dominant scattering mechanism is acoustic phonon

scattering, rS varies between 1.08 for T = 100, to 1.24 for T = 400 K.

The deviation from the traditionally expected value of rS =.1.18 is due

to the contributions of optical phonon modes at the higher temperatures.

Hole-hole collisions also affect the impurity and optical phonon scat-

tering contributions so they become significant even at low temperatures

and dopant densities. At higher values of dopant density, the effects

of hole-hole scattering on the ionized impurity scattering mechanism














1.80



1.60.


1.40




1.20




1.00


0.80




0.60


100 150 200 250 300 350


400


Temperature (K)








Figure 5.1. The mass anisotropy factor rA as a function of temperature
for various impurity dopant densities.


































1015 1016 1017
Impurity Dopant Density (cm-3)


Figure 5.2.


The mass anisotropy factor rA as a function of impurity dopant density for
various temperatures.


0


1.60



1.40


1.20




1.00


0.80



0.60
1014














1.30


100 150 200 250 300 350


400


Temperature (K)








Figure 5.3. The scattering factor rS as a function of temperature for
boron-doped silicon with dopant density as a parameter.







-- -I i i c I rI -T i-'--T 'T 1 I- I ii -I i \ I ii

1.30






400 K

300 K
1.20





200 K



1.10








1 .001 -- | I I 1 i I \ I -I-I 1! I I 1 1 | l I J1 I I 1 i l

1 4 005 117 IL8
1014 1015 1016 1017 1018
Impurity Dopant Density (cm-3)

Figure 5.4. The scattering factor rS as a function of dopant density for boron-doped
silicon with temperature as a parameter.







18 -3
become very noticeable. At NA = 10 cm the highest value of rS is
1.29 for T = 100 K, where the dopant impurities are only about 30 per-

cent ionized [17]. At higher temperatures where the percentage of

ionized impurity atoms is over 80 percent, the effects of hole-hole

scattering bring rS from its traditionally expected value of 1.93 to

1.05 for T = 400 K.

5.5 Hall Mobility and Hall Factor in the Combined Valence Band

Expressions for Hall coefficient, applicable in the case where

holes in more than one band take place in conduction, are given by
Putley [66]. For the case of p-type silicon, assuming no compensation

and operation in the low field region, the Hall coefficient is given

by [66]

3 2
RHiOCi
RH =i=l (5.10)
H 3 2
GCi
{i=1 j

By substituting equations (2.9) and (5.6) through (5.10) into equation

(5.4), the Hall mobility in the combined valence band of silicon can
be expressed by


3 m~.3/2
f j21
i=1 mHi
PH = e 3/2T (5.11)
S Di
i=l mCi







The conductivity mobility for the combined valence band can be expressed

by


3 m* 3/2
ic m* il D i } (5.12)
Dm i=1 !i 1i>


Then using equations (5.11) and (5.12) we can express the Hall factor in
terms of the scattering relocation times and effective masses of the
individual bands by
3 m 3/2
m*3/2 i=l mDi }



m*
i=^1 JHi
rH 3 m* 3/2 (5.13)
Di

Figures 5.5 and 5.6 summarize the results of equation (5.13) as a
function of temperature and dopant density. These figures show that the
Hall factor ranges theoretically between 1.73 and 0.77 for temperatures
between 100 and 400 K and dopant densities between 1014 and 101 cm3.
For temperatures above 200 K, rH becomes less than unity for dopant
15 -3
densities greater than 5.5x10 cm Figures 5.7 and 5.8 show the
theoretically predicted Hall mobility as functions of temperature and
dopant density. These two figures show the results of evaluating equa-
tion (5.11) with the aid of numerical integration, and adjusting the
lattice scattering mobility to give the best fit to values of conduc-
tivity mobility deduced from resistivity measurements.











































100 150 200 250 300 350


Temperature (K)









Figure 5.5. Theoretical Hall factor vs temperature for boron-doped
silicon with dopant density as a parameter.









1.8



1.6 1



1.4

200


1.2 300 K

400 K

1.0



0.8



0.6
1014




Figure 5.6.


Boron Density (cm-3)
Boron Density (cm )


Theoretical Hall factor vs dopant
as a parameter.


density for boron-doped silicon with temperature


S













































100 150 200 250 300 350


400


Temperature (K)


Figure 5.7.


Theoretical Hall mobility as a function of temperature for
boron-doped silicon with dopant density as a parameter.











S I100 K -


150 K

200 K

o 3 250 K

S300 K




400 K
0 2 350 K
---

( -






10 I

1014 1015 1016 1017 1018

Boron Density (cm-3)



Figure 5.8. Theoretical Hall mobility as a function of dopant density for boron-doped silicon
with temperature as a parameter.














CHAPTER VI

EXPERIMENTAL PROCEDURES


6.1 Introduction

Experimental measurements of resistivity, Hall coefficient and

dopant density were made on six silicon wafers, four doped with gallium,

and two doped with indium, in dopant densities ranging from 4.25x1015 to
17 -3
3.46x10 cm These wafers were cut from crystals grown along the

<111> and <1.00> direction. Additional data were obtained from boron-

doped silicon wafers to further verify the adequacy of the theory. The

data were obtained from test,patterns NBS-4 [67] fabricated on the

silicon wafers. This test pattern was designed at the National Bureau

of Standards primarily for use in the evaluation of the resistivity

versus dopant density relation in silicon. Resistivity measurements

were made on four-probe square array resistors and collector Hall effect

resistors, while the net dopant density in the specimens was determined

by the junction C-V method on a gated base-collector diode. Mean values
of resistivity, dopant density and Hall coefficient were determined by

measuring five to eight selected test cells with a standard deviation

in resistivity at 300 K under 5 percent. The following sections

describe the test sample preparation and fabrication procedure, and the

measurement procedures.








6.2 Fabrication Procedure

The overall pattern is fabricated on a square silicon chip 200 mils

on a side where six mask levels are used [68]. The masks were used in

the following sequence: base, emitter, base contact, gate oxide, contact

and metal. Appropriate cleaning procedures (see Appendix A) precede the

.diffusion of impurities, and a negative photoresist process was used in

the masking steps. The base mask delineates regions whose conductivity

type is opposite from that of the collector substrate, and the emitter

mask delineates regions whose conductivity type is the same as that of

the collector substrate. A base region approximately two pm deep is

diffused into the background material; then the emitter region is dif-

fused into the base to a depth of approximately one pm. The base con-

tact mask is used to open windows onto the base region, where an n+

diffusion is made to improve ohmic contact to the base. The gate oxide

mask delineates regions where an oxide layer of closely .controlled

thickness is grown to serve as a gate for MOS devices. After front-side

metallization, a portion of the wafer was separated. This section was.

scribed to provide the Hall effect devices. The remainder of the wafer

was then metallized on the backside and alloyed. After scribing, the

devices were mounted on TO-5 headers, metal contact bonding was made,

and the devices were encapsulated. A layer of ceramic insulating mate-

rial was used to isolate the devices from contact with the header.

Resistivity measurements were then made to select devices for use in

this study.








6.3 .Experimental Measurements

The structures used to evaluate the resistivity of the bulk mate-

rial are the Hall effect resistor and the collector four-probe resistor

[68]. The four-probe resistor has four point contacts arranged in a

square array. The structure (see Appendix A) is fabricated by diffusing

a base over a large area except at the four point contacts which are

protected from the base diffusion by oxide islands. Emitters are dif-

fused at these points in order to make low resistance contacts to the.

collector material. The purpose of the base diffusion is to eliminate

surface currents. The bulk resistivity is determined by forcing a

current, I, between two adjacent probes and measuring the voltage, V,

between the other two probes. The resistivity of the material is deter-

mined from [69]


2?TSV
p SV (6.1)
(2-v2)IC'

where S is the probe spacing and C' is a correction factor dependent on

the ratio of probe spacing to the thickness of the chip [70]. This

correction factor is given by


S 4nw2]2-1/2
C' 1 + 4 (l)n 1 +4nw
2-,2 n=l S2



2 (-1 ) 1 + (6.2)
2-2 n=l S2 2](62


where 'w is the thickness of the chip.

The collector Hall effect resistor is a four-terminal resistor

formed in a square chip 100 mils on a side. Contacts are formed on the





81


four corners by an emitter diffusion (see Appendix A). The resistivity

is calculated from [71]

TWw V
P Tw V (6.3)
p = Xn2 I


where w is the thickness of the chip, and V is the voltage difference

between nearest neighbor contacts for a current, I, passed between the

remaining two contacts. The TO-5 header was mounted in the sample

holder of an Air Products and Chemicals AC-3L CRYO-TIP liquid nitrogen

system. This enabled variation of the sample temperature between 100

and 350 K. The temperature was measured by a chromel vs gold with 0.07

atomic percent iron thermocouple. For temperatures above 350 K, the

sample was placed in a Stratham Temperature Test Chamber.

The structure used for the Hall coefficient measurements is the

collector Hall effect resistor. The Hall coefficient is calculated

from

V Hw
RH V (6.4)
RH BI


where VH is the voltage difference measured between opposite contacts

for a current, I, passed between the remaining two contacts, and B is

the magnetic field density perpendicular to the plane of the chip; thus

the samples are oriented so that the magnetic field is in the crystal

growth direction, <111> for the gallium-doped samples, and <100> for

the indium-doped samples. The Hall mobility is determined from








RH
HH = p (6.5)


where p is determined from resistivity measurements on the Hall and

four-point structures. The magnetic field for the Hall measurements

was provided by a Varian Associates (V3703) six-inch electromagnet with

a current regulated power supply (V-FR2503). The magnetic field

strength was monitored by a Bell 620 gaussmeter with an STB4-0402 probe

with a stated accuracy of 0.1 percent. Data were taken over a tempera-

ture range from 100 to 350 K. The current used in the resistivity and

Hall coefficient measurements was provided by a Keithley 225 current

source capable of accuracy within 0.5 percent of the three-digit

readout. The current was monitored by voltage readings across precision

resistors connected in series with the current-source. These resistors

were part of a Dana-651 current shunt set, accurate to within 0.01 per-

cent. Voltages were measured with a Hewlett Packard 3465A digital

multimeter with a stated accuracy within 0.03 percent of the readout.

Resistivity and Hall coefficient measurements were made in accordance

with ASTM standard procedures [72].

The impurity dopant density was obtained by use of two different

structures: an MOS capacitor, and a base-collector diode. The MOS

capacitor over collector consists of a main gate which is surrounded

by a field plate that overlaps a channel stop which also serves as top

side collector contact [68]. This structure (see Appendix A) is used to

measure the collector dopant density (NA + ND) from the high frequency

C-V deep depletion method [73]. The collector dopant density is deter-

mined by obtaining a dopant profile from C-V measurements by means of









N(x) = 2 AV- (6.6)
eESA'2 AC2


where AV is an incremental, change in the gate voltage, and the measured

capacitance is due to both the oxide and the semiconductor. A self-

consistent check was made on the measurements of collector dopant density

by using the base-collector diode. This structure (see Appendix A)

consists of a base diffused into a collector and a metal field plate to

control the periphery. The field plate overlaps both the base and a

diffused emitter channel stop which also serves as topside collector

[68]. To obtain a correct density profile the field plate is biased at

the flat-band potential [74]. Capacitance-voltage measurements were

taken with a Princeton Applied Research 410 C-V Plotter and a Hewlett

Packard 701OA X-Y Recorder.

From each silicon wafer, eight four-probe resistors, eight Hall

resistors, and eight capacitor-diode chips were selected for encapsu-

lation. These were chosen on the basis of low leakage currents and good

contacts at the metal bonding pads. Measurements were made on each of

the devices and data from the five to eight devices closest to the mean

value of the measurements were then averaged. In this manner we arrived

at representative values of resistivity, Hall coefficient, and dopant

density for each sample. The results of these measurements and compari-

sons with the theory of Chapters III through V are presented in the

next chapter.




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