Citation
The mobility, resistivity and carrier density in p-type silicon doped with boron, gallium and indium

Material Information

Title:
The mobility, resistivity and carrier density in p-type silicon doped with boron, gallium and indium
Creator:
Linares, Luis Carlos, 1944- ( Dissertant )
Li, Sheng-San ( Thesis advisor )
Lindholm, Fredrik A. ( Reviewer )
Kurzweg, Ulrich H. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1979
Language:
English
Physical Description:
xvii, 148 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Average linear density ( jstor )
Boron ( jstor )
Conductivity ( jstor )
Density ( jstor )
Electrical resistivity ( jstor )
Hall effect ( jstor )
Impurities ( jstor )
Ion density ( jstor )
Silicon ( jstor )
Temperature dependence ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Semiconductor doping ( lcsh )
Silicon ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Using the relaxation time approximation and a three-band model (i.e., nonparabolic light-hole band, parabolic heavy-hole and split-off band), a derivation involving the use of the Boltzmann transport theory was applied to obtain expressions for the valence band density-of-states effect if mass, mD, the valence band conductivity effective mass, mC, and the valence band hall effective mass, mH, of holes in p-type silicon. 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 three-band model, theoretical calculations of hole mobility, resistivity, Hall factor and Hall mobility were conducted for p-type silicon doped with boron, gallium, and indium for dopant densities from 10^14 to 10^18cm^-3 and temperatures between 100 and 400 K. Scattering contributions from acoustical and optical phonon, 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 int he calculations. The valence band density-of-states effective mass, mD, was found to vary from 0.6567 mo at 100 K to 0.8265 m o at 400 K, while the valence band conductivity effective mass, mC, increased from a value of 0.3604 mO at 100K to a value of 0.41910 mO at 400 K. The valence band Hall effective mass, mH, varies from 0.2850 mO at 100 K to 0.5273 mO at 400 K. The masses at mD and mC 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 ansotropy 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 form 4.25x10^15 to 9.07x10^17 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 out 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 than 5x10^16 cm^-3. 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.
Thesis:
Thesis (Ph. D.)--University of Florida, 1979.
Bibliography:
Includes bibliographic references (leaves 144-147).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Luis Carlos Linares.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Luis Carlos Linares. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
023424510 ( ALEPH )
07119890 ( OCLC )
AAL5197 ( NOTIS )

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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 supervisory committee for their support and cooperation. In particular, I thank Dr. S. S. Li for his guidance, enthusiasm and professional example-, 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 Mob ility in the Combined Valence Band.


Time


PAGE



iv

vii

xi xv



I


6 8

12

26 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 Resistiv-ity of p-Type Silicon. . . .55

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

5.1 Introduction . . . .62

52The 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. . . . . . . . .10

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


* . . 24 . . . 25


. . . 42


43


. . . 44


Q 0








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

47 Theoretical calculations of resistivity vs dopant
density for boron-doped silicon with temperature
as a parameter. . . . . . . .5

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

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

5.1 The mass anisotropy-factor rpA 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. . . . . . . . . . . . .7

5.4 The scattering factor rsas 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 . . . . . . . . . . . . . .

56 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


77


85 87 88 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. 4.25xi015 cm-3 . . . . . . . . . . . . . . . .

mobility vs temperature for gallium-doped sample. 4.09xi016 cm-3 . . . . . . . . . . . . . . . .

mobility vs temperature for gallium-doped sample. 1,.26xi017 cm 3 . . . . . . . . . . . . . . ,

mobility vs temperature for gallium-doped sample. 3,46xi017 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


A Inverse mass band parameter

A' Area of the base-collector diode

a Defformation potential constant (acoustic phonon scattering)

B Inverse mass band parameter

b Defformation potential constant (optical phonon scattering)

C Inverse mass band parameter

CP Longitudinal sound velocity in silicon

C t Transverse sound velocity in silicon E Energy of holes

e Magnitude of the electronic charge

EA Acceptor impurity energy level
EF Fermi energy level

E V Valence band edge El Binding energy of neutral acceptors

fo The Fermi-Dirac function

F Fermi-Dirac integral of order 1/2

g Ground state degeneracy

1 Plank's constant divided by 2'r

I Current

3 Current density

k Wave vector

ko 0 Boltzmann's constant










m 2 m*
C
* m*
D


m*




NA
m




NN
G






NV Pi pD

rA i RH
rH










RHi.
rHi



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

Ratio of defformation potential constants Y Function of band mass parameters

Ratio of density-of-state effective masses
Yhha Hole-hole reduction factor for acoustic phonon scattering Yhh Hole-hole reduction factor for ionized impurity scattering Y hh0 Hole-hole reduction factor for optical phonon scattering A Energy of spin orbit splitting
Reduced energy (E/k T)

0Variable of integration E2 Variable of integration


E s Relative dielectric constant

Limit of integration Reduced Fermi-energy nIl Scaling factor
0 Spherical coordinate

.0D Debye temperature PC Conductivity mobility in the combined band

UCi Conductivity mobility in band i
Hall mobility in the combined band

Limit of integration defined in Figure 2.1

P Resistivity of holes

Ps Density of silicon

aC Electrical conductivity

aH Hall conductivity

T Total scattering relaxation time

T aci Acoustic phonon scattering relaxation time in band i


xiii








Ionized impurity scattering relaxation time in band i Tij Total interband scattering relaxation time Tii Total intraband scattering relaxation time

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

Adjustable scattering constant

Spherical coordinate














Abstract of Dissertalion 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, m , and the valence band Hall effective mass, m*, of holes in p-type silicon. 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 mobility, resistivity, Hall factor and Hall mobility were conducted for p-type silicon doped with boron, gallium and indium for dopant densities from 1014 to 1018 cm-3 and temperatures between 100 and 400 K. Scattering 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, m*, was found to vary from 0.6567 m0 at 100 K to 0.8265 m0 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 m0 at 400 K. The valence band Hall effective mass, mH, varies from 0.2850 mo at 100 K to 0.5273 mo at 400 K. The masses m* and m* showed little change with dopant density, but m* varied by as muchas 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 4.25xi015 to 9.07xi017 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








than 5xO16 cm3. 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

I NTRODUCT ION


The goal of this study has been to measure and compare with theory the resistivity and Hall mobility of holes in silicon doped with gal lium 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 conductivity 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 a pplied field, and the momentum lost in collisions which tend to randomize the carrier momentum [1]. If the field is expressed in volts per centimeter, 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 dete rmined'by the different types of collisions which the carriers undergo. Coll isions of-carriers with lattice atoms which








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 temperatures and low dopant densities, scattering by lattice phonons is more effective while at low temperatures and high impurity densities, scattering 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 ove-r 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 scattering 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 s.tudy 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 nonparabolicity and all the relevant scattering mechanisms with the exception of hole-hole scattering, but he limited his investigation to dopant densities below 5xlO16 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 withboron 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 functions 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 and gallium 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-detec tor 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 4.25x101 to 9.05x10 . cm. 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 10 18 cm-3 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 mobil ity 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 contribute to the mobility have different temperature and energy dependences, 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 Hal 1 mobility are presented. Fabrication techniques and experimental procedures 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 modeling.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 Hal.l effect in silicon. Including the band nonparabolicity in calculations of relaxation time via the effective mass formulation is a reasonable 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 calculations in silicon.

Lax and Mavroides [20] have derived expressions for density-of-states effective masses m*l and m*2 for the heavy-hole band and the light-hole band, respectively, which lead to the generally accepted and quoted value, m* 0.591 mo. This value, however, can only be considered applicable at 4.2 K, where m* = 0.537 m0 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 densities, 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 curvature 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 effective mass, m*, is the mass of a mobile charge carrier under the application of external electric and magnetic fields. The reason for these particular definitions of effective masses is that the primary application 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 m*, 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 structure of silicon is pr esented in Section 2.2, and in Section 2.3 expressions for m*, m*, 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 independent, but direction-dependent effective mass. The light-hole band








is characterized by holes with an energy and direction-dependent effective mass. These two bands can be described by the E vs k relationship [28]
.-

E~) ___1 (k2 E 24 + C2 (k 2 k2 + k 2 k2 + k 2k 2 (21
E(k) _ Ak2 [B x y x z y z } (2.1)



where A, B, and C are the experimentally determined inverse mass band
2 + k2 + k)2 and the upper sign is associated with parameters, k z(k k

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 = 0 by an energy A = 0.044 eV. [32], and is characterized by an effective 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)
2m


where












0 and are the spherical coordinates, E V is the top of the valence band, and


I + 1 y[sin 4 6(cos 4 + sin 4 Cos 4 0 - 2/31 (2.4)


with


Y C 2 /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 m* 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
2
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 FermiDirac 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 V = 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.


2
K (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, mi, is defined from the relationship

4 2,fkoTm*i 3/2

pi 4 2 k Di F1 2( ) (2.6)


where

I
Fl/2(n) = I + exp (- ) (2.7)


a =(EV - E)/koT, n (Ev EF)/koT, 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]


Ji = gjkEk + CjkQEkfH + Ojk.Zm'k HkHm + .


(2.8)








where Ek, H, Hm are the electrical and magnetic field components and the a's represent single-energy-surface conductivity coefficients. The first coefficient in equation (2.8) is the zero-magnetic field electrical 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 conductivity coefficient, aC, to define the conductivity effective mass m*i, by the relationship


e2
OCi lli Pi m (2.9)


and the Hall effect coefficient, aH5 to define the Hall mobility effective mass by means of [37]


e3
aHi ' 123i i (2.10)



To solve for m i, mi, and mi equations (2.6), (2.9) and (2.10) are equated to the following expressions for pi, Gjk' and jkZ:
1

Pi 3 f f0(k)d3k (2.11)
47


e2 fo 3E 3E 3k
ejk T 3 a d (2.12)




4jk T31 f4E fTj Dp q k pq








where f is the Fermi-Dirac distribution function and cpq is the permutation 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 mi, mti, 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 mi and mi we also require a procedure for evaluating and <2 > 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


M1 m� [f(-y)]2/3 (2.14)



mo0 f(-Y) (2.15)
mCl = - P fland

mwher 0 fi(_�eie (2.16)


where y is defined 1n equation (2.5). In these equations


f(y) = (1 + 0.05y + 0.01635y 2 + 0.000908Y 3 +








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


+ 0.00091959y4 +


and


f2 ().= (1 0.01667 + 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

2mo3/2 f/k0T
2)*3/2 0 f(+�) f C_':d c




A-'3/2 f /poT
(A-B') 3/2 /k T jx~ (.7








2m 0 T2F3/2dc
C 2' f exp(c)

S+Y /koT, C12 dc f (-Y)nl Co ide1

f(+'y) ,rk0 ex p-(d + -/"___ ex.(1)


e'poc) (A-B')1/2 /koT (�I



(2.18)



(A+B') /2f2(+Y) f/k�T T223/2exp(-c)dc +


(A-B')1/2f2(-Y)n1 f 0 T 2 2 13/2exp(-I )dc1 /koT 1 H2co E 3/2d c F f(+ ) f /koT C d c
e x, epT-) _B 1 )3/2-exp(c) V o L(A+' 032
f (-Y)Tql o 001 l .1(2 19
(A-B')3/2 i/ e x pT Fl) y


where c1 c - A/3ko0T, C,= A/3, q, exp(-A/3ko0T) and A and 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 nonparabolicity of the light-hole band introduces a dependence on the scattering 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


E=Ev - A- A (2.20)


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 02A~
m exp (-3(2.21)
D3 A3




A 3/2
fT3F2 exp(-E2)d 2
0

m f T33/2exp(_)d
m* 0 0 (2.23)
H 3 A - TA3 C2 e x p (- 2 )d 2
MI f _3 2 C3/2 exp( _ 2)d 2




where c2= c - A/k0T.

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= P1 + P2 + P3 (2.24)


thus


m*.= [(m* )3/2 + (m 2 + ,m*3/2 2/3 (2.25)
D Dl Dm2 Dm3J


This combined effective mass is the mass corresponding to the densityof-states of an effective single equivalent parabolic valence band. This concept is useful in calculations where the effective density-ofstates at different temperatures can be calculated from one m*.

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 effective 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 =CI + 'C2+ 'C3 (2.26)


and


H = Hl + H2 + aH3 (2.27)


respectively.








Substituting equations (2.9) and (2.10) into equations (2.26) and (2.27) it follows that


rfl~ 3/2 M <2 m23/2 i <> [m* )3/2 ~~1
m=C m* m m-- --- c D) C 2 t m * mC3)
(2.28)

and


3/2 .T 2> 3/2 Dm21 1 +f3oD313/ I
.* FD2 2 -m 2 2 2 l
H <[2> D <'T > m*J 2 1
m 2 <2.2 J H m m J m*32
H2 H3
(2.29)


Equations (2.25), (2.28) and (2.29) were evaluated numerically as functions of temperature and acceptor doping density for p-type silicon. Values of the band parameters, IAI = 4.27, IBI = 0.63 and CI = 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 rigorous 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 explicit 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 m*l. The temperature

dependence of m3 is more pronounced since here we also have the
D3
effects of energy displacement at T = 0. The temperature dependence due to nonparabolicity is very apparent in the shape of the m*2 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.


I00


150 200 250 300 350 400
Temperature (K)


Temperature masses nl1


dependence of the density-of-state effective mi2 and m*3, in the individual bands, and the


combined density-of-states mass m* of holes in silicon.
14 - 3D N1A = 10 cm








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 density equal to 1014 cm3. 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 k0T, 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 m* can be attributed mainly to the nonparabolicity of the light-hole band. In the temperature range from 100 to 400 K, m* increases from 0.2850 to 0.5273 m. The slight temperature dependence of m*l and m*l is due to the explicit temperature effect and results in increases of 7.7 percent and 3.76 percent in the m*l and respectively. A larger temperature variation occurs in the case of the split-off band because of the additional effects of the energy displacement at k = 0.

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

density and temperature. For T 2 100 K, m* varies less than 10 percent 14 18 -3
in the dopant density range from 10 to 10 cm . 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



S0.40



4-)
4.


4-)

U

0 0.20



0.10
100





Figure 2.3.


Temperature (K)


Temperature dependence of the conductivity effective masses M1., m2 and m*3 in'the individual bands, and the combined conductivity effective mass m* of holes in silicon. NA = 1014 cm3





" 23


100 150 200 250 300 350


Temperature (K)


Figure 2.4.


Temperature dependence of the Hall effective masses, ml m*2 and m*3 in the individual bands, and the combined Hall effective mass m* of holes in silicon. N = 1014 cm-3.


0.60


0.50


(0
4



4
LU





0


300 K


400 K


350 K


200 K .


250 K 150 K 100 K


0.50


0.40


0.30


10 15


10 1.6 N A (cm- 3


10 17


10 18


Figure 2.5.


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





















o ~ 150 K




0.30 - 10 "K.






0.20



1014 1015 1016 017 1018

NA(cm- )


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 hea.vily-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 mi and m* are in excellent agreement with those of Barber [25]. We have extended Barber's work to the calculations of m* and m* 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 m0, is somewhat 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 m* 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 calculations of effective masses were achieved through more rigorous mathematical 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 m* could increase by as much as 31 percent in the range of temperatures from 100 (M*
c
0.510 M ) to 300 K. This is i.n reasonable agreement with our calculated
0
percentage increase in m* in the same temperature range (33 percent), but it is impossible to compare our calculations with Hauge's experimental 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 approximation of a constant effective mass seems to be inadequate to describe transport properties of holes in silicon above 100 K. There is a substantial 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 effective 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 l14 to 3xlO18 cm3. This calculation is limited to applications in conductivity 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 resistivity 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 conductivity mobility is the mobility associated with the conductivity expression, cr 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 unwrapping 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 P=0, (b) relaxation time indepe ndent 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 s-ubstitutes 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 mechani sms are dominant in certain temperature and dopant density regimes, but in some cases two or more may be interacting simultaneously. Thus in calculating 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-riole scattering also plays an important role in determining 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 acc omplished by evaluating the mobility separately in the heavy-hole band, the light-hole band, and the split-off band considering all appropriate scattering mechanisms. Th e 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


f 3/2- dF = f (3.2)
1 I 3/2 {[;f� d



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 temperature 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 + rDi Ti T i # j; i = 1,2; j = 1,2 (3.3)
lj
�D ij+,


where








1TT22
TI 2 T21


and


Tji=Taci -I +T -I +T Ii- + T-N l (3.5)


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


-3 = I+ U3 + [ 13- + TN3] (3.6)


Only transitions between the light- and heavy-hole band are considered; the relaxation time Tji takes into account a transition from band i to band j; and Taci' Toi' TIi and T Ni 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 replacing 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 silicon. 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 m*l 32 L1 (2) + 3 L ()
acl TX 11 ij 11


C2
2 2 (2) + Yij3 (1)] T 3/21/2
112 + T T (3.7)




and


-1 m2 L (2) + y 3 L2(1) +
Tac2 Tx L22 i 22



c z B21T-(2) + Y 3 T(I T3/2Cu/2 (3.8)

C 2 22ij 22 9








for intraband scattering, while


-1 - -I mi3/2
- (2)+
t ij Tji ij
x


C 2 2Tij(2)] } y3/21/2 (3.9)

C t


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


ac3 T T T (3.10)



In these equations


k3/2a2 m3/2
1 p0 am0 (3.11)
T x 42_7TI4P C 2z


yi. Di D' b/a, a and b are valence band acoustic deformation

potential constants in the Picus and Bir [44] notation, ps is the density, CP and Ct are the longitudinal and transverse sound velocities in silicon and Lij and Tij are functions of and Yij defined in [43].


3.4 Optical Phonon Scattering

Optical phonon scattering, while negligible at very low temperatures, 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]


-1 m* 3/ Dg~l r (no+l)r _i
T -1 = i T112 - D 1~/2
oi - x D.


+ 'D } i = 1,2,3 (3.12)




where 0D is the Debye temperature, no = (exp(OD/T)-l)- is the phonon distribution function, and W is a constant which determines the relative coupling strength of the holes to the optical phonon mode compared to the acoustical phonon mode

D 2fi 2 C 2
W o (3.13)
2ko a 0D2

2
.where D0 is the optical deformation potential constant. The first term
0
in the brackets of equation (3.12) corresponds to optical phonon emmission and is relevant only when this is energetically possible (>e D/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 charged impurities 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]
4
-e 4NAG(bi)
-l A2m i)I/2 s2koT) 3/2 i 1,2,3 (3.14)



where


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



and,

247T m*iFs(koT)2
b1 e2h2p,



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

ND = 0.


3.6 Neutral Impurity Scattering

Scattering by neutral impurities in semiconductors has been considered 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









T E-1 s 2 Oi'N i =1,2,3 (3.1)
Ni = 1m*. N'
~m~e Di


where NN is the density of'neutral impurities and m* 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 threedimensional square well to estimate the influence of a weakly-bound state on the scattering. In this case the relaxation time is given by.




-l_ 232rfi (k T) 1/2m-* {2 / + k T �12 1 T,2;3 (3.18)

where 2

E l.136 x 10-1 - - (3.19)
M [O
0


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 highe r temperatures, the influence of neutral impurity scattering can extend over a wide range of temperatures.


3.7 Effect of Hole-Hole ScatteringThe expressions thus far presented for scattering relaxation time neglect the effect of hole-hole scattering. Although hole-hole scattering 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 reduction factor, Y hh' 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




<'T hh > = f(3.20)

f �"/2T_ f0d


where f0 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




hho �/{~.dc x f C 32T_ [ dc.
{F 3/2a ojd]









_hh 1 (3.21a)
hh

0
for optical phonon scattering, and Yhh0' 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 Yhh = 97T/32 0.88 [17] in a certain range of impurity concentration. The exact relationship (Yhh = 1.0004 - 4.013378 x 109NA 1015 NA 3 10 17) 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 1 [- - expP2] (3.22)
NA

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

Thus the overal 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 scattering. 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

m~l 3/2 l- 13/2 m*3 3/2
iD= lLm + P2{m 3 + 13 {m] (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 s 1013cm-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 and 5x1013cm-3 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
PC 2/C t2

0D PS

s

Tx W m
0
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

F0
sec K3/2



kg joule-sec joules/K coul









































-3
Boron Density (cm)


Figure 3.1.


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


0


!
w





�
(2)



05


102












































































10 18


Gallium Density (cm- )


Figure 3.2.


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


0


(-.3
C)
C.,)


E (-.3



-I-)



~0
0


C)

0


1101





0


101017 1018

Indium Density (cm- )


Figure 3.3.


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











































100


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.











































150 200 250 300 350 400


Temperature (K)


Figure 3.5.


The calculated hole mobility vs temperature for galliumdoped silicon with dopant density as a parameter.









































150 200 250 300 350


400


Temperature (K)


Figure 3.6.


The calculated hole mobility vs temperature for indiumdoped'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, T x was found by fitting the mobility to experimental data in the lattice-stattering-limited range. Our value of T x 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 6.024x10 8 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


2k Tm* 3/2
p = h2 0 F1(n) (4.1)


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 N 2(2 Tr m* k T/h2)3/2 is the effective density o'f valence band
whr V D( 0 k

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 N0 = 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[ kTFJ



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[- k-J (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 energyEA 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 3xIO18 cm-3. Fo r impurities with deeper activation energies, it is also expected that at some impurity concentration, the impurity activation energy should become a function of the impurity concentration. However, in the case of gallium 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 concentration, 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 temperature as a parameter for 100 T 5 400 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


































2K



0
10 14


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




















































1015 1016


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.


0


4-)
U S.
Ga








C_


1014


1018





0


I 00i


10 18


10.14


10 15


10 17


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.








temperatures to achieve total ionization of impurities. The deionization 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)
epICp


where PC 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 temperature, for 1014 - N 1018 cm-3 and 100 T ! 400 K. The results are displayed in Figures 4.4 through 4.9. In the calculations of resistivity in silicon doped with gallium and indium, as was done for conductivity mobility, it was assumed that boron impurities were also present.
1313 -3
Boron densities of 10l3 and 5xlO1 cm 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.


10 1 100


10-1





10-2


1 00;









































I0-I1




10

100








Figure 4.5.


Temperature (K)


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





























101





I00


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.






























I00


10-1





10-2


1015 1016 1017

Boron Density (cm- )


Figure 4.7.


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


18 10













































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.


I0-I


10-2

1014


1015


1018






















101 100


10-I


1018


1016 10 17
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 HaynesSchokley drift method, wherein the drift of charge carriers in a known electric field is measured. However, the assumption, made when these experiments were ini tiated, 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 differ-s 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 g reater than 10O8 cm-2 gauss/volt-sec [60]) the Hall coefficient is simply related to the
carrier concentration by [61]


RH1(51
H pe








and thus the conductivity mobility and the Hall mobility are equal. Although the high field limit simplifies use of the Hall mobility considerably, excessively high magnetic fields can cause problems due to the quantization of the hole orbits in a magnetic field [I]. The quantization of the particle motion in a magnetic field will create Landau levels within the band. The Landau levels will modify the density of states in the valence band which could affect the interpretation 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)



For nondegenerate, spherically symmetric bands, it can be shown that [59] rH 1 1, and that

<,r2>
rH - r2(5)



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) - E-1/2 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 be equal 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 overestimated by the assumption of unity Hall factor. A more complete theoretical treatment of the Hall factor can be undertaken by considering 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= C 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)
CCi


Thus by substituting equations (2.9) and (2.10) into equation.(5.5) the Hall coefficient can be expressed as


R rHi (5.6)
Hi pie


where









imi2 <2>
r Hi [mHJ[> (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

M*2
rAi : i(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 silicon. 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 6xlO15 cm3. At 300 K, rA is less than unity even for dopant densities as low as 1014 cm-3


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 traditionally expected variation between 37r/8 = 1.18 and 315ir/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 r S =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 scattering 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 r A as a function of temperature
for various impurity dopant densities.






































1015 1016 17

Impurity Dopant Density (cm- )


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 r S as a function of temperature for
boron-doped silicon with dopant density as a parameter.







I I"�I i i riT-T-TiYr1 . I ii I| i I i i I 1
1.30








/300 K i
1.20




- /I
200 K "1 .0 0 . 1 "

1o4 1o5 16 17 1o8
101 101 101 101 101
-3
Impurity Dopant Density (cm-) 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 percent 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
RHilCi
RH = i=l (5.10)

.= i


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*i3/2
i i


PH e 3 3/2
i=l mCi








The conductivity mobility for the combined valence band can be expressed by

3 e* m3/2

i c 2{i " } (5.12)
Do i 'Ci

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 *3/2
Mm3/2 i32 }
Hi
rH 3 m*3/2 (5.13)


i=l m i 2


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 1018 cm3 For temperatures above 200 K, rH becomes less than unity for dopant 15 - 3
densities greater than 5.5xi0 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 equation (5.11) with the aid of numerical integration, and adjusting the lattice scattering mobility to give the best fit to values of conductivity 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.














1.6 F




1.4




1.2 300 K

400 K

1.0 0.8




0.61~
10 14




Figure 5.6.


Boron Density (cm-3)


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













10'
- 150 K

200 K








010
L



10- I 2 1350 K










'1014 l15 1016 l17 1018
3
Boron Density (cm-)



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 4.25xi015 to

3.46xi017 cm-. These wafers were cut from crystals grown along the <111> and <100> direction. Additional data were obtained from borondoped 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 diffused into the base to a depth of approximately one im. The base contact 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 material 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 material 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 diffused 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 determined from [69]


2SV (6.1)
(2-v72)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

4 4n 2w2 -1/2
C' - 1 + - (_I)n 1 + 4nw
- 2-v2 n=l S2


2 -2 nO )n [1 2n2w2]-I/2
2 -,/ n=l S (62


where 'w is the thickness of the chip.

The collector Hall effect resistor is a four-terminal 'esistor

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]


Trw V (3
Xn2 1 (6.3)


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 - . (6.4)


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
H (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 temperature 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 O.Ol percent. 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 determined by obtaining a dopant profile from C-V measurements by means of









W~x 2 2 AV- (6.6)
ec A2 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 selfconsistent 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 7010A X-Y Recorder.

From each silicon wafer, eight four-probe resistors, eight Hall resistors, and eight capacitor-diode chips were selected for encapsulati-on. 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 comparisons wit h the theory of.Chapters III through V are presented in the next chapter.




Full Text
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