THE MOBILITY, RESISTIVITY AND CARRIER DENSITY
IN pTYPE 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 indiumdoped 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. 735741 and the National Science Foundation
Grant No. ENG 7681828.
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 HoleHole 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 pType Silicon . . . . . 55
V THE HALL FACTOR IN pTYPE 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 densityofstate
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
borondoped silicon with temperature as a parameter
3.2 The calculated hole mobility vs dopant density for
galliumdoped silicon with temperature as a parameter
3,3 The calculated hole mobility vs dopant density for
indiumdoped silicon with temperature as a parameter
3.4 The calculated hole mobility vs temperature for
borondoped silicon with dopant density as a parameter
3.5 The calculated hole mobility vs temperature for
galliumdoped silicon with dopant density as a
parameter . . . . . . . . . .
3.6, The calculated hole mobility vs temperature for
indiumdoped 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 borondoped silicon with dopant density as a
parameter. . . . . . . . .. .. . 56
4.5 Theoretical calculations of resistivity vs temperature
for galliumdoped silicon with dopant density as a
parameter . . . . . . . . . . . . 57
4.6 Theoretical calculations of resistivity vs temperature
for indiumdoped silicon with dopant density as a
parameter . . . . . . . . .. .. .. .. .58
4.7 Theoretical calculations of resistivity vs dopant
density for borondoped silicon with temperature
as a parameter . . . . . . . ... .... 59
4.8 Theoretical calculations of resistivity vs dopant
density for galliumdoped silicon with temperature
as a parameter . . . . . . . .... ... .. 60
4.9 Theoretical calculations of resistivity vs dopant
density for indiumdoped 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 borondoped silicon with dopant density as a
parameter . . . . . . . . ... .. .. . .70
5.4 The scattering factor rS.as a function of dopant
density for borondoped silicon with temperature
as a parameter .. . . . . . . . . .. .71
viii
FIGURE
5.5 Theoretical Hall factor vs temperature for
borondoped silicon with dopant density as
a parameter . .. . . . . . . . .
5.6 Theoretical Hall factor vs dopant density for
borondoped silicon with temperature as a parameter
5.7 Theoretical Hall mobility as a function of temperature
for borondoped silicon with dopant density as a
parameter . .
5.8 Theoretical Hall mobility as a function of dopant
density for borondoped silicon with temperature
as a parameter . . . . . . . . .
7.1 Hole mobility vs hole density for borondoped
silicon at 300 K . . . . . . . . .
7.2 Resistivity vs dopant density for borondoped
silicon at 300 K . . . . . . . . .
7.3 Resistivity vs dopant density for gallium and
indiumdoped silicon at 300 K . . . . . .
7.4 Resistivity vs temperature for the.borondoped
silicon samples . . . . . . . . .
7.5 Resistivity vs temperature for the galliumdoped
silicon samples . . . . . . .
7.6 Resistivity vs temperature for the indiumdoped
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 galliumdoped sample.
4.25x1015 cm 3 . . . . . . . . .
mobility vs temperature for galliumdoped sample.
4.09x1016 cm 3 . . . . . . . .
mobility vs temperature for galliumdoped sample.
1.26x1017 c3
1.26x 01 cm . . . . . . . . .
mobility vs temperature for galliumdoped sample.
3.46x1017 cm3
mobility vs temperature for indiumdoped sample.
16 3
4.64x10 cm
ix
PAGE
. .
FIGURE PAGE
7.12 Hall mobility vs temperature for indiumdoped sample.
NA = 6.44x1016 cm3 . ................ .99
7.13 Hall factor vs dopant density for ptype silicon
at 300 K . . . . . . . . . . ... . .. 100
KEY TO SYMBOLS
Inverse mass band parameter
Area of the basecollector 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 FermiDirac function
FermiDirac 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
Heavyhole mass at 4.2 K
Lighthole mass at 4.2 K
Conductivity effective mass in the combined band
Conductivity effective mass in band i
Densityofstate effective mass in the combined band
Densityofstate 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 coefficientin 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 densityofstate effective masses
Yhh Holehole reduction factor for acoustic phonon scattering
Yhh Holehole reduction factor for ionized impurity scattering
Yhh Holehole 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 Fermienergy
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 pTYPE SILICON DOPED WITH BORON, GALLIUM AND INDIUM
By
Luis Carlos Linares
August 1979
Chairman: ShengSan Li
Major Department: Electrical Engineering
Using the relaxation time approximation and a threeband model
(i.e., nonparabolic lighthole band, parabolic heavyhole and splitoff
bands), a derivation involving the use of the Boltzmann transport theory
was applied to obtain expressions for the valence band densityofstates
effective mass, m*, the valence band conductivity effective mass, mC,
and the valence band Hall effective mass, m*, of holes in ptype 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 threeband model, theoretical calculations of hole mobil
ity, resistivity, Hall factor and Hall mobility were conducted for
ptype 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 holehole scattering
on the various scattering mechanisms, and the nonparabolicity of the
valence band were also taken into account in the calculations. The
valence band densityofstates 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 cm3, 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 ptype 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 borondoped 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 holehole 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
ptype silicon have been conducted [116]. 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 ptype silicon in
which lattice scattering dominates; Braggins [1] considered nonparaboli
city and all the relevant scattering mechanisms with the exception of
holehole scattering, but he limited his investigation to dopant densities
below 5x1016 cm3 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
ptype silicon over a wide range of temperatures and dopant densities.
This improved model was applied to the case of borondoped silicon with
great success [17]. The improvement in the theory consisted mainly of
the inclusion of holehole 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 [1416], most of the research in ptype 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 photodetector devices. Curves of resistivity and mobility as func
tions of dopant density [2,3] have been applied to characterizing
borondoped 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
onchip 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 photodetector 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 valenceband structure and derivation of expressions for
thetemperature 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 holehole
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 densityofstates
effective masses m*D and m*2 for the heavyhole band and the lighthole
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 splitoff 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 FermiDirac
statistics.and a simplified model of the valence band structure for
silicon, Barber [25] obtained an expression for the densityofstates
effective mass, m*, which is temperature and holedensity dependent.
Barber, however, did not apply the nonparabolic model of the valence
band to the study of conductivity or Hall effective mass in ptype
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 splitoff band
and the temperature variation of the band curvature.
In this study, the expressions for densityofstates effective mass,
conductivity effective mass, and Hall effective mass of holes are derived
based on the following definitions. The densityofstates 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
ptype silicon over a wide range of temperature and dopant density.
Since the crystal structure of silicon has cubic symmetry, the ohmic
mobility and the lowfield 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 heavyhole and
lighthole bands, degenerate at k = 0, and a third band displaced down
in energy at T = 0 by spin orbit coupling.
The heavyhole band is characterized by holes with an energy inde
pendent, but directiondependent effective mass. The lighthole band
is characterized by holes with an energy and directiondependent effec
tive mass. These two bands can be described by the E vs k relationship
[28]
__(2 +2 (k2k2 k2k2 k2k2
E(k) = 2o 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 lighthole band, while the lower sign is associated
with the holes in the heavyhole 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 lighthole 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 IIIV compounds, is reasonable
in the case of Ge and Si [27,31]. The splitoff 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 heavyhole 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 lighthole band is considered parabolic with a constant slope
corresponding to the value of m* at 4.2 K. For higher energies the
lighthole band is assumed to take on approximately the same slope as
that of the heavyhole band, but remains separated from the heavyhole
band by A/3 eV [27]. The extrapolation of these two constant slopes
creates the kink in the lighthole band at 0.02 eV. Because of the
change in slope, the lighthole band has an energyvarying effective
mass and in general can only be described in terms of partial Fermi
Dirac integrals [25]. Although the splitoff 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 heavyhole 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
heavyhole band based on Kane's model [27] is reasonable. Other studies
[35,36] support the validity ofthis 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 densityofstates 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(cn) (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 heavyhole,
lighthole, and splitoff 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 singleenergysurface conductivity coefficients. The
first coefficient in equation (2.8) is the zeromagnetic 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 (mi)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 FermiDirac 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 carrierconcentration dependent. Although equations
(2.6) and (2.11) through (2.13) are expressed in terms of FermiDirac
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 HeavyHole Band
In this band, the effective masses are given by
Mm* [f(Y)]2/3 (2.14)
mD1 (AB')
mI A f(Y) (2.15)
ml TA(P fl( y
and
me =(AB 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 heavyhole 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 LightHole Band
In the lighthole band, as modeled by Figure 2.1, the effective
masses of holes are obtained in terms of partial FermiDirac 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)
(ABI) 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() (AB')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() (AB')1/2 /k T exp(e)
(2.18)
S(A+B')1/2 f /k T 223/2exp(E)dE +
(AB')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
(AB')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 FermiDirac integrals and equations (2.6),
(2.9') and (2.10) were expressed in terms of complete FermiDirac
integrals, the dependence on T does not cancel out. Thus the nonpara
bolicity of the lighthole band introduces a dependence on the scatter
ing relaxation time. The scattering relaxation time is discussed in
Chapter III.
2.3.3 The SplitOff Band
Although the splitoff 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 splitoff 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 splitoff 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 densityofstate 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
ofstates of an effective single equivalent parabolic valence band.
This concept is useful in calculations where the effective densityof
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 densityofstates 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 ptype 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 densityofstate effective
m*2 and m*3, in the individual bands, and the
D'2 "D3'
combined densityofstates 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 lighthole 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 lighthole 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 lighthole 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
splitoff 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+
uI
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(cm3)
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 temperaturedependent 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
heavilydoped ptype silicon. Cyclotronresonance studies conducted by
Hensel and Feher [22] show that when carrier heating populates deeper
regions'of the lighthole 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 densityofstates
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 ptype 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 splitoff 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 curvefitting type of procedure.
The experimental values of densityofstates effective masses of
holes in ptype 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. Magnetokerr 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 lighthole band, and a smaller, though not
negligible, contribution due to the explicit temperature dependence and
the effects of the splitoff band. The validity of this model for the
calculation of densityofstates effective mass has been well established
[25]. Barber [25] has shown that when the temperaturedependent 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 ptype
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 cm3. 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 easilymeasured 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'ptype 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. Holehole 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 heavyhole
band, the lighthole band, and the splitoff band considering all
appropriate scattering mechanisms. The overall mobility is then
evaluated as a weighted average of the singleband 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 FermiDirac 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 ptype 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 lightholes (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 splitoff band is given by
o3 = [+ 3 + T13 + TN3 1 (3.6)
Only transitions between the light and heavyhole 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 ptype 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 PhononScattering
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 m1 L (2) + 3 (1)
acl Tx 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 splitoff 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 T2.3
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 cutoff 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 electronhydrogen impurity scattering by using a three
dimensional square well to estimate the influence of a weaklybound
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 1019 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 HoleHole Scattering
The expressions thus far presented for scattering relaxation time
neglect the effect of holehole scattering. Although holehole 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,
holehole 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 holehole scattering on the scattering relaxation time is a function
of the energy dependence of the relaxation time. The holehole reduc
tion factor, Yhh, can be derived by means of a classical formulation
introduced by Keyes [54]. When holehole collisions are much more
frequent than holeacceptor 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 FermiDirac distribution function. On the other hand,
if holehole collisions are neglected, the average relaxation time is
given by equation (3.2).
Thus the holehole 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)
/ arj dI]
I U ^~r .
Yhh 1 (3.21a)
Yhh 
for optical phonon scattering, and Yhho, the holehole 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 oneparticlelike approximation from
the HartreeFock theory, have shown that by inclusion of holehole
scattering, the BrooksHerring [48,49] formula is reduced by a factor
which can be expressed in closed form as
Yhh ; I exp2] (3.22)
N
where NAis the ionized acceptor density and p' is the screening hole
density. In the case of neutral impurity scattering, holehole 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 holehole 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 singleband
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 31, 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 3cm3 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 1000Qcm, it was assumed that boron densities of 1013
13 3
and 5x10 cm existed in the gallium and indiumdoped 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 indiumdoped 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 ptype silicon. The figures also show that for the case of the
Table 31. 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 1010
0.244
9.1 x 1031
6.25 x 1034
1.38 x 1023
1.6 x 1019
Unit
meV
eV
eV
K
kg/m3
0
sec K3/2
kg
joulesec
joules/K
coul
Boron Density (cm3)
Boron Density (cm )
Figure 3.1.
The calculated hole mobility vs dopant density for borondoped silicon with
temperature as a parameter.
0
U
U
Q
a)
0
102
1018
Gallium Density (cm3)
Figure 3.2.
The calculated hole mobility vs dopant density for galliumdoped 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 (cm3)
Figure 3.3.
The calculated hole mobility vs dopant density for indiumdoped silicon with
temperature as a parameter.
150 200 250 300 350
400
Temperature (K)
Figure 3.4.
The calculated hole mobility vs temperature for borondoped
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 lightlydoped 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 latticescatteringlimited 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 FermiDirac 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 densityofstates 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 ptype 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 splitoff 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 (cm3)
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 (cm3)
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 (cm3)
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 pType Silicon
The resistivity of ptype 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 cm3 were assumed to exist in the
gallium and indiumdoped 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 ptype 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
borondoped silicon with dopant density as a parameter.
101
100
101
102
100.
10
10
102
100
Figure 4.5.
Temperature (K)
Theoretical calculations of resistivity vs temperature for
galliumdoped silicon with dopant density as a parameter.
101
100
101
100
Figure 4.6.
150 200 250 300 350
400
Theoretical calculations of resistivity vs temperature for
indiumdoped silicon with dopant density as a parameter.
100
101
102
1015 1016 1017
Boron Density (cm3)
Figure 4.7.
Theoretical calculations of resistivity vs dopant density
for borondoped silicon with temperature as a parameter.
1018
1016 1017
Gallium Density (cm3)
Figure 4.8.
Theoretical calculations of resistivity vs dopant density
for galliumdoped silicon with temperature as a parameter.
101
102
1014
1015
1018
101
100
101
1018
1016 1017
Indium Density (cm3)
Figure 4.9.
Theoretical calculations of resistivity vs dopant density
for indiumdoped silicon with temperature as a parameter.
CHAPTER V
THE HALL FACTOR IN pTYPE 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 ntype sample is the same as when they constitute the
majority carriers, is invalid in view of carriercarrier 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 highfield limit (when the
product of mobility and magnetic induction becomes greater than 108 cm2
gauss/voltsec [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 wavefunctions 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 ptype 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 holehole 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 galliumdoped 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 CV 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 holehole 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.
Holehole 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 holehole 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 (cm3)
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
borondoped 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 II 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 (cm3)
Figure 5.4. The scattering factor rS as a function of dopant density for borondoped
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 holehole
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 ptype 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 borondoped
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 (cm3)
Boron Density (cm )
Theoretical Hall factor vs dopant
as a parameter.
density for borondoped 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
borondoped 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 (cm3)
Figure 5.8. Theoretical Hall mobility as a function of dopant density for borondoped 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 NBS4 [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 fourprobe square array resistors and collector Hall effect
resistors, while the net dopant density in the specimens was determined
by the junction CV method on a gated basecollector 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 frontside
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 TO5 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 fourprobe resistor
[68]. The fourprobe 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)
(2v2)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]21/2
C' 1 + 4 (l)n 1 +4nw
2,2 n=l S2
2 (1 ) 1 + (6.2)
22 n=l S2 2](62
where 'w is the thickness of the chip.
The collector Hall effect resistor is a fourterminal 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 TO5 header was mounted in the sample
holder of an Air Products and Chemicals AC3L CRYOTIP 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 galliumdoped samples, and <100> for
the indiumdoped samples. The Hall mobility is determined from
RH
HH = p (6.5)
where p is determined from resistivity measurements on the Hall and
fourpoint structures. The magnetic field for the Hall measurements
was provided by a Varian Associates (V3703) sixinch electromagnet with
a current regulated power supply (VFR2503). The magnetic field
strength was monitored by a Bell 620 gaussmeter with an STB40402 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 threedigit
readout. The current was monitored by voltage readings across precision
resistors connected in series with the currentsource. These resistors
were part of a Dana651 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 basecollector 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
CV deep depletion method [73]. The collector dopant density is deter
mined by obtaining a dopant profile from CV 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 basecollector 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 flatband potential [74]. Capacitancevoltage measurements were
taken with a Princeton Applied Research 410 CV Plotter and a Hewlett
Packard 701OA XY Recorder.
From each silicon wafer, eight fourprobe resistors, eight Hall
resistors, and eight capacitordiode 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.
