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This is to certify that the

thesis entitled

Measurement of Thermal Conductivity of Boron
Doped CVD Diamond Films as a Function
of Electrical Resistivity

presented by

Andrew Sayers

has been accepted towards fulﬁllment
of the requirements for

Master of Science Mechanical Engineering

degree in

 

 

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MSU le An Afflnnetlve ActIoNEquel Opportunlty Inetitulon

 

MEASUREMENT OF THERMAL CONDUCTIVITY OF BORON
DOPED CVD DIAMOND FILMS AS A FUNCTION
OF ELECTRICAL RESISTIVITY

By

Andrew Sayers

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1996

ABSTRACT

MEASUREMENT OF THERMAL CONDUCTIVITY OF
BORON DOPED DIAMOND FILMS AS A FUNCTION
OF ELECTRICAL RESISTIVITY

By

Andrew Sayers

A novel experimental technique has been used to determine thermal conductivity
of boron doped diamond ﬁlms. The method allows for electrical resistance heating for a
range of resistivities, therefore possessing the capability to estimate thermal
conductivities for different doping levels. Boron doped diamond ﬁlms were deposited
using the hot ﬁlament technique. Boron powder precisely placed during the diamond
deposition process contributed to diamond growth with varying resistivities from sample
to sample (and within each sample). The resistance of each sample was measured using
the four probe method. Samples were chemically etched to obtain a 3mm diameter of
free standing diamond ﬁlm. The samples were then prepared. energized. and monitored
using an infrared imaging system. Temperature distribution across the 3mm diamond
window was extracted and used to estimate the ﬁlm’s thermal conductivity. Results
show that thermal conductivity decrease as doping level increase. The thermal
conductivity values range from 205 to 975 W/m K, while the uncertainty range from i] 0
to i65. The samples’ resistivities. which were used to quantify the level of doping. were
measured in the region where temperatures were acquired and varied from 1.63 (2 cm to

68.4 Q cm.

ACKNOWLEDGEMENTS

I would personally like to thank the three people who made this whole experience
possible, Dr. Beck, Dr. Mcgrath, and Dr. Somerton. Each of these individuals has given
me a great deal of guidance and support at Michigan State University. The time spent

working with each of them has not only enhance me intellectually, but also personally.

Many thanks go to Dr. James Beck for all his support. I consider Dr. Beck a
wonderful individual as well as a great scholar. His patience and support has always been
there whenever it was needed. The time spent in his classroom and laboratory was very

rewarding.

I would also like to extend my gratitude to Dr. John Mcgrath for his expertise in
performing the experiments and overall guidance throughout. Dr. Mcgrath brilliance and
experience was crucial in every phase of the project. and his critical reviews made the

ﬁnal version possible.

I also thank Dr. Craig Somerton for making all this possible. Dr. Somerton efforts
in recruiting and retaining my services cannot go unnoticed or unappreciated. The

experience gained while working under his supervision was highly rewarding.

Throughout this project there were many people who contributed their time and
effort in helping. While most of their names are mentioned in the text, one individual
went beyond what was necessary. I extend my thanks to Gwo-shi Yang for his effort in

depositing the ﬁlms.

Last, but not least, was my family. This would not be possible without their
continued support. Although they are a few hundred miles away, I could not survive
without their assistance. To my mom Gloria, my dad Julian, brothers Greg and Tom, and

sister Debbie thank you very much.

 

LIST OF TABLES
LIST OF FIGURES

LIST OF SYMBOLS

1 Introduction

2 Background/Literature Review
2.1 Atomic structure of diamond
2.2 Mechanism of diamond growth
2.3 Diamond as a semiconductor

2.4 Determination of thermal conductivity

3 Mathematical Model
3.1 Heat conduction model
3.2 Radial heat ﬂow model
3.3 Simpliﬁed model
3.3.1 Radiation consideration

3.3.2 Convection consideration

viii

xi

27

28

4.1

4.2

4.3

4.4

4.5

4.6

5.1

5.3

5.4

5.5

5.6

5.7

6.1

6.2

Sample Design
Wafer preparation
Mask

Diamond deposition
Etching

Sample preparation

Gold deposition

Experimental Method/'1‘ echnique
Experimental setup

Infrared thermography
Determination of emissivity
Measurement of resistivity
Measurement of ﬁlm thickness

Temperature acquisition

48

48

51

55

64

67

86

Data processing and parameter estimation of thermal Conductivity 90

Results and Discussion >

\\

Thermal conductivity as a function of electrical resistivity

Uncertainty analysis

vi

7 Summary and Conclusions

8 Recommendation for Future Work

Appendix

Bibliography

vii

123

126

128

183

Table 2.1
Table 2.2
Table 5.1
Table 5.2
Table 5.3
Table 5.4
Table 5.5
Table 5.6
Table 5.7
Table 5.8
Table 5.9

Table 5.10
Table 5.11
Table 5.12

Table 6.1
Table 6.2
Table 6.3

Table 6.4
Table 6.5
Table 6.6
Table 6.7
Table 6.8

LIST OF TABLES

Round Robin Testing

Yang’s Resistance Measurements

Emissivity of Black Paint

Emissivity of Diamond (Test #1)

Emissivity of Diamond (Test #2)

Emissivity of Diamond (Test #3)

Emissivity of Diamond (Test #4)

Resistivity of Samples

Global Resistivity using DMM

Thickness Measurements using RST—Plus

Thickness Measurements using ESEM

Thickness Measurement using Sloan Dektak IIA
Summary of Thickness Measurements

Uncertainty and Standard Deviation of Each Instrument
Estimated Doping Level for all Samples

Average Resistivity and Standard Deviation

Estimates of Thermal Conductivity with Average and Standard
Deviation

Uncertainty in Thermal Conductivity

Parameter Contribution to Uncertainty in Thermal Conductivity

Middle Resistivity, Uncertainty, and Percent Uncertainty
Average Resistivity, Uncertainty, and Percent Uncertainty
Parameter Contribution to Uncertainty in Resistivity

viii

12
6O
61
62
62
63
66
67
71
82
85
85
86
100
103

103
114
118
119
120
120

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6

Figure 2.7
Figure 2.8

Figure 3.1
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14
Figure 5.15

LIST OF FIGURES

Cubic Structure of Diamond

CVD Growth Environment

Dynamic Balance Model

Yang’s Experimental Setup

Thermal Conductivity as a Function of Temperature
Thermal Conductivity as a Function of Methane to Hydrogen
Ratio

Thermal Conductivity as a Function of Methane Concentration
Local Thermal Conductivity Parallel to the Surface as a Function
of Temperature

Radial Heat Flow Model

Schematic of HF CVD System

Photograph of HFCVD System

Silicon Wafer with Mask

Side view of the Etched Diamond Sample

Top view of the Etched Diamond Sample

Conducting Strip

Sample Setup

Photograph of Sample Viewed from the Diamond Side
Photograph of the Sample Viewed from the Silicon Side
Photograph of the Emscope SC 500 Sputter Coater
Schematic Diagram of the Experimental Setup
Photograph of the Data Acquisition System

Photograph Model 600L (Scanner and Control Unit)
Optical Path of the Scanner

Photograph of Lens

Black Plate with Heater Attached

Black Plate at Room Temperature

Temperature Distribution of Heated Black Plate

Black Plate with Attached Sample

The Four Probe Method

Photograph of the Four Probe Instrument

Sloan Dektak IIA

Step Height using RST-Plus (Sample #17)

Step Height using RST-Plus (Sample #3)

Corner Proﬁle of Thickness (Sample #17)

Hr-‘NDOOQ
LAN

19

21
26
34
35
37
41
41
42
43
43
44
46
49
50
52
53
55
56
58
58
59
64
65
68
7o
70
72

Figure 5.16
Figure 5.17
Figure 5.18
Figure 5.19

Figure 5.20

Figure 5.21
Figure 5.22
Figure 5.23
Figure 5.24
Figure 5.25
Figure 5.26
Figure 5.27
Figure 5.28
Figure 5.29
Figure 5.30
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9

Figure 6.10
Figure 6.11
Figure 6.12

Figure 6.13
Figure 6.14

Figure 6.15

Corner Proﬁle of Thickness (Sample #3)
Bullseye at the Center for Spherical Shape (Sample #3)
Focal Plane of LSM
Thickness Measurement on the left edge of Sample #3
using LSM -
Thickness Measurement on the right edge of Sample #3
using LSM
Thickness Measurement Sample #4 using Newview 100
Thickness Measurement Sample #5 using Newview 100
Oblique Plot of Surface for Sample #5
Oblique Plot of Surface for Sample #4
Surface Proﬁle of Sample #4
Surface Proﬁle of Sample #5
Thickness Measurement (Test #3)
Thickness Measurement (Test #7)
Thickness Measurement (Test #9)
Adjustable Clamp with Sample
Measured versus Calculated Temperature (Sample #17)
Residuals (Sample #17)
Sequential Estimation of Thermal Conductivity (Sample #17)
Thermal Conductivity as a function of Average Resistivity
Irvin Curve for Diamond
Thermal Conductivity as a function of Middle Resistivity
Combine Plot of Measured Thermal Conductivity
Graebner et a1. results showing effects of each type of defect
Curve Fit of Thermal Conductivity as a Function of
Middle Resistivity
Curve Fit of Thermal Conductivity as a Function of
Average Resistivity
Comparison between Calculated Uncertainty and
Measured Uncertainty
Measured Uncertainty Relative to Mean Thermal
Conductivity Estimates
Sensitivity of Thermal Conductivity to Thickness of Diamond
Experimental Results plotted as a Function of Average
Resistivity with Error Bars
Experimental Results plotted as a Function of Middle
Resistivity with Error Bars

72
73
74

75

76
78
78
79
79
80
8O
83
84
84
87
94
95
96
97
99
101
102
106

109

110

115

116
117

121

121

7F>f""U7<
'7:-

‘5’
o

D.

3

awnﬁamampwwaD—ir
('3

'0

LIST OF SYMBOLS

Thermal Conductivity
Power

Length

Area

Boltzmann Constant
Planck Constant
Temperature
Temperature difference, Debye Temperature
Thermal Conductivity
Thickness

Emissivity

Surrounding Temperature
Minimum Temperature
Temperature Amplitude
Quantity of Heat

Heating Frequency
Speciﬁc Heat

q,,‘qy,qz Heat Flow in the x,y,z Direction

Resistivity, Density

Internal Heat Generation

Convection Coefﬁcient

Radial Position

Convective Fin Term

Modiﬁed Bessel Function of the First Kind of Order Zero
Modiﬁed Bessel Function of the Second Kind of Order Zero
Heat Flow due to Radiation

Radius

Sheet Resistivity

Voltage

Current

Measued Temperature

Calculated Temperature

Thickness of Diamond

Thickness of Gold

Resistance

xi

Boundary Temperature
Least Square Function
Phonon Frequency
Phonon velocity
Constants

Uncertainty

xii

CHAPTER 1

INTRODUCTION

Thin ﬁlms have been known to mankind for hundreds and possibly thousands of
years. Examples would include the layers of paint used by prehistoric man to decorate
his caves, and more recently some of the decorative glazes that were applied to primitive
pottery. However, in the space of a few years, thin ﬁlms have been elevated in status
from relative obscurity to a ﬁrmly established technology which plays an important role
in many industrial processes. Thin ﬁlms as a specialized technology have come to refer
to layers deposited by some type of atom by atom process rather than to layers produced
by laying down relatively large particles to form a truly continuous layer. The type of
thin ﬁlms discussed here will be restricted to those deposited in some type of vacuum or
low pressure environment. These techniques have been known for some time, and are
centered around three major methods: vacuum evaporation, sputtering, and chemical
vapor deposition (Maissel, 1973). In the following work, two of these techniques
(chemical vapor deposition and sputtering) are utilized in an experimental technique for

the determination of the thermal conductivity of thin diamond ﬁlms.

Diamond is a material which has captivated the imagination of scientists and
engineers for centuries, not only for its rare jewel status, but also because of its hardness,

chemical composition (inert) and high thermal conductivity. Because of diamond’s

unique physical properties, it has been proposed for application as a heat sink, wear
resistant coating, cutting tool, and protective coating for infrared optics in harsh
environments (May 1995). The use of diamond in such applications was hindered due to
its limited abundance in nature and unsuccessful attempts by scientist and engineers to
synthesize it in the laboratory. Diamond synthesis has attracted attention after it was
discovered in 1797 that diamond is a crystalline form of carbon. Initially, synthesizing
was attempted at high pressure since that diamond is a very dense form of carbon. In
1955, Bundy and coworkers at General Electric succeeded in the synthesis of diamond
(Nassau, 1993). Their success was achieved at very high pressure and temperature,
therefore limiting widespread production. However, recent developments in Chemical
Vapor Deposition (CVD) methods has transformed diamond from a rare gem to an

engineering material which can be produced easily and used rather extensively.

As mentioned above, one of the most attractive properties of diamond is its high
thermal conductivity. Type IIa single crystal diamond has exhibited thermal conductivity
values as high as 2200 W/m K, more than ﬁve times that of copper (400 W/m K).
However, the degree to which CVD diamond ﬁlm compares to single crystal diamond
depends signiﬁcantly on the preparation conditions. The conditions for synthesizing
CVD diamonds ﬁlms and its relationship to thermal conductivity has yet to be understood
fully. Preliminary results of thermal conductivity have shown variation by a factor of 3

to 5 (Graebner, 1994).

One preparation condition of major importance is boron doping of diamond
during synthesis. Boron allows the diamond crystals to become charge carriers, and the
added impurities can greatly affect the thermal properties. Therefore, the thermal
conductivity of diamond ﬁlms must be determined as a function of the doping level of
each ﬁlm. Before this can be accomplished, a method to measure the doping level of
each ﬁlm must be determined. The boron used during synthesis causes the diamond
crystals to become charge carriers, therefore allowing for electrical conductance. The
doping level can be measured by relating it to the electrical resistance or resistivity of the

ﬁlm.

 

CHAPTER 2

BACKGROUND/LITERATURE REVIEW

One of the most attractive properties of diamond is its very high thermal
conductivity, which in single crystals can be as high as 2200 W/m K (Graebner,1992).
Polycrystalline diamond created by Chemical Vapor Deposition (CVD) has also shown
high thermal conductivity. However it is not as large as that of its single crystal
counterpart. Early reports for thermal conductivity of CVD diamond ﬁlms indicate
variation by factors of 3 to 5 from sample to sample (Graebner, 1994). While different
experimental techniques has been used to determine thermal conductivity, it is presumed
that this variation is due to the conditions under which the diamond ﬁlms were fabricated
rather than the technique used. However, measurements of a single sample by different
researchers has also shown variation. Table 2.1 shows results of tests performed on one

sample by different researcher.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sample Sinku-Riko Norton Wurzburg GE AT&T
(W/m K) (W/m K) (W/m K) (W/m K) (W/m K)

LD«X 15.5 14.1 25.6

LD-E 2.6-4.4 3.6 6.6

SQ-E 2.6-4.0 4.7 5.1
SQ-T 5.2 8.0 14.0
SQ-F 5.9 9.2 12.3
SQ-FB 4.1 6.6 6.7
SQ—X 8.0 10.3 18.0

 

Table 2.1 Round robin testing

 

The results in Table 2.1 shows that measurements which should be comparable can
actually differ by a factor of two (or more). In this chapter, a basic understanding of the
atomic structure responsible for diamond’s high thermal conductivity is presented, along
with the mechanism necessary for diamond growth and the various techniques (and
preparation conditions) used to experimentally determine the thermal properties. The
principles and background presented in this chapter are essential to understand the doping

process, and its affect on the thermal conductivity of diamond ﬁlms.

2.1 ATOMIC STRUCTURE OF DIAMOND

As in any crystalline solid, the atoms in diamond are linked together in a simple
and regular way. Diamond is exclusively a face-centered cubic structure with atoms
bonded together in strong sigma type covalent bonds. The cubic structure is shown in

Figure 2.1.

   

Figure 2.1 Cubic structure of diamond (W ilks, 1991)

As shown in Figure 2.1, the structure consists of eight comer atoms, six face-center, and
four other atoms. Each carbon atom is tetrahedrally coordinated to four carbon atoms via
sigma bonds emanating from Sp3 hybrid atomic orbitals (Wilks, 1991). Within the cube
are four atoms shown as open circles each of which is linked by four bonds to one of the
black atoms around it. This arrangement repeats itself through the diamond. The atoms
shown as open circles also lie on the points of another face-centered cubic lattice similar
and parallel, but somewhat displaced from the ﬁrst lattice. Therefore, the complete
diamond lattice may be described as consisting of two interpenetrating face centered
cubic lattices (Wilks, 1991). It is this structure and strong molecular bonding which

gives diamond its extreme hardness and high thermal conductivity.

In polycrystalline diamond ﬁlms made by CVD, various preparation conditions
can contribute to impurities and geometric defects in the atomic structure. These
anomalies can have a signiﬁcant affect on the physical properties (including thermal
conductivity). Therefore, in order to characterize CVD diamond ﬁlm properties, the
inﬂuence of impurities and geometric defects on the thermal conductivity must be

understood for different preparation conditions.

2.2 MECHANISM OF DIAMOND GROWTH

The growth of diamond from the vapor phase in a reactor begins with nucleation,
the initial formation of a crystal, followed by epitaxial growth, the deposition of
additional carbon on the surface to increase bulk dimensions. While there are many CVD
techniques which have successfully produced diamond, the basic principles necessary for

an operative system are given below.

The CVD growth environment can be quite complex and varied, as shown in

Figure 2.2.

 

RBACI'AN’I‘S

II
H2+CH4

   
 

    

ACHVA'HON
H2 r. beet 2H
CH4 + H——CH3 + H2
\\ : /
\L/

FLOW AND REACTION

     
  

 

““‘ "”"'"“ I min-ion 1"”

b—ﬂ

  

115-111333.983. '-

Figure 2.2 CVD Growth environment (Pehrsson, 1993)

The reactants, methane and hydrogen, enter a high temperature or energetic region in
which the gas is activated by a plasma, hot ﬁlament, or a combustion ﬂame front. The
species participating in this reaction are transported by forced ﬂow, diffusion, and

convection throughout the reactor, eventually reaching the substrate (Pehrsson, 1993).

In order for nucleation to proceed, an incipient nucleus must form and be stable
long enough for lattice growth to occur. The development of a critical nucleus (one large
enough to survive and grow rather than decay) depends on the system free energy
changes accompanying particle formation under growth conditions. The change in the
system free energy must be negative for the process to occur, where the system includes

the substrate, growing particles, and the gas-phase reservoir of species contributing to

lattice growth (Pehrsson, 1993). Therefore, the substrate and its pretreatment prior to

growth have a large inﬂuence on nucleation and the initial stages of growth.

Epitaxial growth of diamond can be explained using various macroscopic models.
One such model which is simple to explain is the dynamic balance model. In this model,
the growth environment is viewed as a dynamic balance or kinetic competition between
carbon deposition and etching processes for other forms of carbon , Figure 2.3 (Pehrsson,

1993).

Dynamic Balance Model

 
   
     
   
 

Non-equilibriwn Gas
WM II C. I

-I.Ilg
‘clkiloli’th....I

 

 

Cello:
- f c... in ca. '
as -
.—-—-—-—-
Diamond Amer. Carbon Graphite
ey’ e : 8p?

 

Deposited Solid Carbon

Figure 2.3 Dynamic balance model (Pehrsson, 1993)

Carbon supersaturation provides the driving force for the condensation of various forms

of solid carbon including diamond, while atomic hydrogen provides a reactive

10

environment for the removal or suppression of the non-diamond carbon by etching of
solid carbons and /or converting surface sz bonded sites into Sp3 bonds. While this
model does not identify the detailed gaseous or surface mechanism leading to diamond

growth, it explains general results common to many growth environments (Pehrsson,

1993)

2.3 DIAMOND AS A SEMICONDUCTOR

In order for diamond to become a semiconductor, it must be doped with a material
such as boron during synthesis. Boron allows the diamond crystals to become charge
carriers, therefore allowing electrical conductivity in an otherwise insulating material
(Herr, 1993). Boron was chosen as a suitable dopant because from a geometric and
energetic standpoint, boron is probably the only element that substitutionally dopes
diamond without signiﬁcantly distorting its lattice structure. It can easily be incorporated
into polycrystalline ﬁlms during chemical vapor deposition from either a gas, powder, or
solid source (Das, 1993). The ease with which boron allows for the semiconducting

characteristic to be introduced into diamond makes doping an emerging technology.

The process which explains how the presence of boron leads to an electrical
conductivity in diamond must be investigated on an atomic level. Consider a diamond in
which a small number of carbon atoms have been replaced substitutionally by atoms of

boron. Boron lies next to carbon in the periodic table of the elements so that the atoms

are of quite similar size but have only three electrons to share with the four surrounding
carbon atoms. There is the possibility of electrons from the full valence band moving to
the boron sites to bond with adjacent carbon atoms left unbounded. The boron therefore
acts as an acceptor site, the energy of the extra electron being somewhat higher than the
energy of the top of the valence band. Once electrons have moved up to the acceptor
level, the valence band is no longer completely full. Therefore the electrons in the band
can be accelerated without leaving the band, and carry a current which is proportional to
the number of missing electrons or holes. Note that the diamond will only conduct if
electrons in the valence band have enough energy to move up to the higher levels

associated with boron atoms (Wilks, 1991 ).

While it is fairly easy to dope diamond using boron, it is rather difficult to control
the level of doping. One method presently used to determine the level of doping is the
electrical resistance of the diamond ﬁlm. Since highly doped ﬁlms will conduct
electricity better (lower resistance), measurement of the electrical resistance for a ﬁlm is a
simple and an effective way to determine the amount of boron or level of doping present
in each ﬁlm. While the electrical resistance is easy to measure, it has been shown that
boron doped diamond possesses a great deal of variability of resistance over relatively
small area. In work conducted by Yang (1996) at Michigan State University, the
resistance was investigated over a small area (approximately 2 cm) and the measured

values are shown in Table 2.2. The local regions are

 

Table 2.2 Yang’s resistance measurements (Yang, 1996)
represented by R1, R2, etc., and the dimensions corresponding to each region is shown .
The data shows that the resistance increases as distance from R1 increase, except for the
resistance at R9, which is greater than R10. This is evidence that boron doping using the
method describe here varies randomly. The lengths, widths, and thicknesses are for the
local area which was considered in each case. A diagram showing the setup and region

investigated is also presented (Figure 2.4).

2 2

ﬁﬁiﬁﬁﬁﬁ

q u- .mmmmm‘
' ,
. - . ’0‘ _,_., ~ g - ..

E.

E
1E
.E

3
3.:

 

Figure 2.4 Yang’s experimental setup (Yang, 1996)

The above information shows that boron doped diamond ﬁlms can have resistances which
vary by factors of 5 over a distance of approximately 2 cm. While the resistances in the
ﬁlms used in the present work have less variability (factor of 2 over 1.5 cm), it should be
noted that the spatial distribution of resistance is not uniform in the boron doped diamond

ﬁlms used here.

2.4 DETERMINATION OF THERMAL CONDUCTIVITY

Because of diamond’s fast thermal response, preparation conditions, and
microstructural size, experimental determination of thermal properties is often difficult.
For this reason, most techniques presently being used to detemrine thermal conductivity
involve either special sample design or complicated and expensive equipment setup.
Although complicated, many researchers have successfully studied diamond ﬁlms, and
preliminary results for the thermal conductivity has ranged from about 50 to 2200 W/m
K. This enormous range of values is primarily due to the preparation conditions which
contributed to diamond growth. Many researchers have shown that preparation
conditions greatly affect the quality of diamond produced. Therefore, in order for
diamond to be properly used in many industrial applications, the preparation conditions
and their relationship to thermal properties must be thoroughly understood. The two
most common methods to grow diamond, the hot ﬁlament chemical vapor deposition
(HF CVD) method and the microwave plasma chemical vapor deposition (MPCVD)

method, have proven to exert little inﬂuence on the thermal properties (Graebner, 1994).

However, the preparation conditions (doping, temperature, pressure, gas mixture content,
etc.) can inﬂuence the properties of the diamond signiﬁcantly. The diamond’s quality
ultimately dictates the magnitude of the thermal conductivity. In this section, a
discussion of the various conditions used to grow diamond, and what affect each

condition has on the measured thermal conductivity is presented.

Morelli, Beetz, and Perry (1988) investigated the thermal conductivity of diamond
ﬁlms as a function of temperature. In their work, two diamond ﬁlms with dimension
10mm by 5mm were grown using the hot ﬁlament chemical vapor deposition (HFCVD)
method with chamber pressure of approximately 9 Torr and methane concentration of
about 0.18%. The thermal conductivity was measured using a steady state probe
technique. The setup consisted of thermally anchoring one end of the sample to a copper
clamp which was then exposed to a cold reservoir (closed cycle helium refrigerator). A
lightweight strain gage heater was then attached to the other end using conductive silver
paint. This heater was used to generate a temperature difference, AT, across the ﬁlm. For

a heater power, P, the thermal conductivity, K, of the ﬁlm is given by

P L
K -— (EX?) (2.1)

where A is the cross sectional area of the ﬁlm and L is the distance across the ﬁlm where
AT is generated. Morelli et al. attached a type B thermocouple (chromel-constantan) to
the surface of the ﬁlm. The thermal conductivity was measured over a temperature range

from 10 to 300 K. The measured values for thermal conductivity over the temperature

range studied were smaller in magnitude relative to “bulk” single crystal diamond. In
order to understand the behavior of their samples, Morelli et a1. attempted to ﬁt their data

to a standard Debye-type expression for the thermal conductivity of a crystal given by

k T 4 x
(5)373 circa—c4

k x'e
(ex _1) (2.2)

K:
2n2v

 

Using this model, it was concluded that the lower values for thermal conductivity were
due to scattering processes acting in the ﬁlms. These processes acting on the phonons
were subsequently quantiﬁed to determine the role of each process on the value of
thermal conductivity. Figure 2.5 shows their results along with comparison data from

Berman et al.

10C

 

K (Wen-"(-5

0.1

 

 

 

 

IO 30 1
TIM no 300

Figure 2.5 Thermal conductivity as a function of temperature (Morelli et al., 1988)

16

For diamond, the processes considered included scattering by crystal boundaries. other
phonons via umklapp processes, and point defects such as isotopes and impurities in the

lattice.

One work which speciﬁcally addresses thermal conductivity as a function of the
preparation condition was that by Ono, Baba, and Funamato (1986). In their
investigation, the thermal conductivity was measured for different methane
concentrations during synthesis. In the experimental setup, a thin sample was held by
heated supports at both ends in a vacuum. The ﬁont and rear surfaces of the sample were
coated with black paint so the temperature distribution along the length can be measured
by a thermography system through an optical window. Since the sample was in a
vacuum, heat transfer by convection was eliminated. Therefore, conduction and radiation
were the primary modes of heat transfer. For steady state, the governing differential
equation is

moi—f) = 25,604 — T3) (2.3)

where A = thermal conductivity, d = thickness, 8 = emissivity, and o = Boltzmann

constant. Aﬁer a few simplifying assumptions, the solution is given by

(T? -T..‘)

T = —”4—T—(cosh(kx —1))+ T (2.4)

3 m
m

where TI“ is the minimum temperature of the sample at x=0 and the constant

I
k= (8ah‘,0'7:3 / My , therefore thermal conductivity is given by

A 88h.,0'7:3x2
’ d{cosh"[4733(T- 2.3/(T; - Ti) — 11V

 

(2.5)

The experiment was conducted on free standing polycrystalline diamond ﬁlms
7pm to 30um thick deposited by microwave CVD under a pressure of 4 kPa and a
methane concentration ranging from 0.1% to 3.0% volume ratio of methane to hydrogen.
The results showing thermal conductivity as a function of methane to hydrogen

concentration ratios are shown in Figure 2.6

 

 

 

 

1 0 r
000 O _ Diamond '
ﬁ ’ ", lTypelbl
2’- '
E 0. —Diomond I
E I' I'- (Type I I "
A 500- ‘ A -
- 93 = ,3 .
r \\ 4
c . . . . “““ -o> ------ c-
,” ins 1.0 20 so
01 03

CHL/HZ lvol%)

Figure 2.6 Thermal conductivity as a function of methane to hydrogen ratio
(Ono et al., 1986)
In looking at the results, it should be noted that the effect of varying the thickness on the
thermal conductivity was not addressed. Previous reports have shown that thermal

conductivity varies with sample thickness. Therefore, the results presented by Ono et al.

l8

possess uncertainty with respect to the variation of both thickness and methane to

hydrogen concentration ratios.

Similar work based on methane concentration was performed by Baba, Aikawa,
and Shahata (1991). In their work, diamond ﬁlms were prepared by the hot ﬁlament
CVD technique with varied CH4/I-Iz ratio, and thermal conductivity was subsequently
estimated as a function of CH, concentration. The AC Calorimetric method was used to
measure the thermal diffusivity, which estimated the thermal conductivity using
published values of density and speciﬁc heat. In the AC Calorimetric method, one end of
the sample is periodically heated by a halogen lamp and the temperature amplitude (AC
temperature) at a point a distance X from the heating area is monitored with a
thermocouple attached directly to the sample surface. The relation between AC

temperature and distance x is given by

_ q _ Bf.
1n|T,(_.|_1n(4Hde) (D)x (2.6)

 

where q is the quantity of heat absorbed by the sample, f is the heating frequency, and C,

d, and D are speciﬁc heat, thickness, and thermal diﬁusivity of the sample, respectively.

For methane concentration which varies from 1% to 5%, the thermal conductivity

varied from 200 to 1200 W/m K. A graph showing the results is given in Figure 2.7.

 

14004

I
I
I

1200 -
1000 L
800 r
600 I-
400 r

200 I

 

 

 

 

o i 2 3 4 5
CH4 Concentration (‘16)

Figure 2.7 Thermal conductivity as a function of methane concentration
(Baba et al., 1991)
The results agreed with those of Ono et al. except for the value of thermal conductivity
(plotted on the y axis) at 1%. The authors suggested that the difference may be due to the
difference in the deposition technique and the ﬁlm qualities, such as grain size,

impurities, defects, etc.

In order to quantify the values measured for the thermal conductivity of CVD
diamond ﬁlms, Graebner et al. (1994) compared isotropically enriched polycrystalline
diamond to that of “gem quality single crystal” diamond. Studies of the thermal
conductivity in enriched synthetic single crystal diamond have shown an increase in

thermal conductivity as the [3C concentration is reduced from its natural abundance of

20

1.1% to 0.07%. The presence of '3C causes point defects which increases phonon-
phonon scattering, thereby adversely affecting the thermal conductivity. To understand
this effect more thoroughly, the authors prepared thick ﬁlms (650nm) using 12C enhanced

methane.

The thermal conductivity was measured using a DC technique, and it was
discovered that the enriched ﬁlms have thermal conductivities of about 2180 W/m K
parallel to the surface and 2600 W/m K perpendicular to the surface. The value of 2600
W/m K exceeds the highest reported data for natural abundance gem quality diamond
(2400-2500 W/m K). Their results proved that enriched CVD ﬁlms can ultimately reduce
point defect scattering of phonons to less than those present in natural abundance

diamond.

In another work by Graebner, Jin, Herb, and Gardinier (1992), the thermal
conductivity of CVD diamond samples of various thicknesses were measured over a
temperature range from 5-400K. It was discovered that thermal conductivity is
anisotropic as well as having a gradient with respect to height above the substrate. In
order to investigate thermal conductivity as a function of thickness, ﬁve samples were
created under similar conditions with the exception of deposition time (and therefore
thickness). The microwave enhance CVD method was used with pressures of 100 Torr
and temperatures between 800 and 900 C. The thermal conductivity was measured using

a DC technique. This technique uses thin-ﬁlm heaters deposited on the diamond surface

21

and ﬁne-wire thermocouples attached with silver epoxy in the standard heated bar
conﬁguration. A second heater near thermal ground permits the detection of and

correction for any heat loss by radiation.

The measurement technique used averages thermal conductivity over the
thickness of the sample. A local conductivity is obtained by subtracting the conductance
of any sample of thickness, Z, from the conductance of the next thicker sample and
divides by the difference in thickness to obtain a conductivity average over only the upper
part of the thicker sample. The local values of thermal conductivity at height, Z, above

the substrate is given in Figure 2.8.

m , . 0 ~ I - e I . T - ‘ . I : I - . a I 1' I I

 

KP MCI?!" K")

 

 

 

 

 

 

o L 1 L L l ' ‘ ' L ' ‘ ‘ . I ' ‘ I J 1 l
400 -50 0 50 100

T (“Cl

 

Figure 2.8 Local thermal conductivity parallel to sample surface as a
function of temperature (Graebner et al., 1992)

22

These values indicate that thermal conductivity increases with distance from the

substrate.

While there are many manufacturing methods (HF CVD, MPCVD, etc.) and
conditions which successfully produce diamond growth, the variation of thermal
conductivity is signiﬁcantly dependent on the speciﬁc conditions for which the specimen
was made. The doping of diamond ﬁlms is just one condition which is not understood,
and therefore doping is the main focus of the present work. Boron doped diamond is an
emerging technology which deserves a great deal of attention for future application as

semiconductors.

CHAPTER 3

MATHEMATICAL MODEL

Before any experiments can be performed to estimate thermal conductivity, an
accurate model must be developed. Without such a model the thermal properties cannot
be estimated, and without an accurate account of all heat transfer modes, any results will
be meaningless. In this chapter, the radial heat ﬂow model is developed, along with some

of the simplifying assumptions used in generating an experimental setup.

3.1 CONDUCTION MODEL

In a solid body that contains variations of temperature, heat ﬂow proceeds from a
region of high temperature to a region of low temperature. The term heat ﬂow is the rate
of energy (joules/sec) associated with the vibration energy of atoms and molecules in the
body, and heat ﬂux is the rate of heat ﬂow per unit area at any point in the body. While
heat ﬂux cannot be directly measured, it can be deduced from the temperature distribution
inside a solid body if the relationship between temperature and heat ﬂux is thoroughly
understood (Beck, 1992). The relationship between heat ﬂux q and temperature T is

given by Fourier’s law which states that

24

51°03!) = —k(F.t)VT(F,z) (3.1)

where k is the effective thermal conductivity, and the minus sign is inserted in accordance
with the second law of thermodynamics. In general, heat ﬂux is a vector quantity, and in

rectangular coordinates the components are given by

_E

8’ E!
I
94%)

qx=_ — qy=

To properly model a heat conduction problem, a solution of the heat conduction
equation (also called the energy equation) must be derived from the general integral form
of the energy equation (Beck, 1992).

07‘
a'th (3.2)

j(zj.ﬁ)dA + jg(r,t)dV = IpCp

07
where q = —k F?— and F is a vector

In the general equation, the ﬁrst term on the left hand side represents the net rate of heat
ﬂow in, the second term represents the rate of internal heat generation, and the third term
represents the rate of heat storage within the system. The general equation accounts for

all heat ﬂow processes within a system.

25

3.2 THE RADIAL HEAT FLOW MODEL

The radial heat ﬂow model was chosen because of its analytic simplicity and
small uncertainty in estimation of the thermal conductivity (Herr, 1993). The governing

differential equation for the radial heat ﬂow includes the following assumptions

1) heat ﬂow dissipated by conduction

2) radiation neglected

3) no energy storage (steady state condition)
4) isothermal in the z-direction (qz=0)

5) diamond is isotropic in all direction

which simpliﬁes the heat conduction equation to

d2T de 2h
drz +FEr-_F(T-T”)+g=0 (3.3)

 

k

a schematic diagram showing the radial heat conduction model is shown in Figure 3.1

26

 

 

Sillc on

 

 

 

Figure 3.1 Radial Heat Flow Model

The boundary conditions are
37. = 0 and TIM, = Constant
& r=0

The assumption of constant temperature at the boundary is made due to the silicon
substrate that remains at the edges of the sample, and the fact that the boundary
temperature can be measured directly. The silicon is more than 50 times as thick as the
diamond ﬁlm and has a thermal conductivity of 150 W/m k so that it acts as a thermal
heat sink. The large thermal mass at the edges allows the temperature at this boundary to
remain at a lower constant value compared to the exposed surface of the ﬁlm at steady

state.

The general solution for the radial heat ﬂow is described by:

27

T(r)= C,Io(mjr)+C2K0(mjr)+—ng-+Tw (3.4)
m
1

Application of the boundary conditions yields

T(r)= g. (1———1°(m’r)
mj k 10(m/b)

 

)+ T... (3.5)

2
where g = volumetic energy generated, mf = 75 , b = radius, T,o = surrounding

temperature, r = radial position, and k = thermal conductivity. Equation 3.5 is the
solution for the temperature distribution for radial heat ﬂow with internal heat generation,
natural convection losses on both sides, and a constant temperature boundary condition at

r=b.
3.3 SIMPLIFIED MODEL

The experimental model can be simpliﬁed if other modes of heat transfer can be

neglected. This section will show that convection and radiation effects can be neglected.

3.3.1 Radiation consideration
Because of the relatively small temperature rise used experimentally (about 10 degrees

above room temperature), the effects of radiation can be neglected. The validity of this

28

assumption can be demonstrated by comparing the heat transfer due to radiation with the

total heat generated within the sample. Heat transfer by radiation is described by

q... = 2A£0(T“ - T..“) (3.6)

where A = 717‘

for r=1.5x10'3 m, A=7.069x10°6 m2, 0' = 5.67 x 10'8 W/m2 K4, T=305 K, T5298 K,
and F062,

qmd=3.88 x 10'4 w

This value for radiation is small when compared with the heat generated (P=V2/R).

Typical values are V=1 volts and R=1.5 0, therefore P=0.667. This analysis shows that

qmd _ 3.88 x 10'4
qmw ’ 0.667

 

= 5.8 x 10‘4 W . Therefore, heat loss by radiation accounts for about

half of 1% of the heat generated.

3.3.2 Natural convection consideration

Because of the small area and small temperature difference, the heat loss due to natural
convection can also be neglected. If the governing differential equation is divided by g/k,
the non-dimensional convection term becomes -2h0/g8. This parameter represents the

fraction of the total heat lost by convection to the total heat generated (Herr, 1993).

29

Substituting the radial heat ﬂow solution evaluated at r=0 into the non-dimensional

convective term yields

 

l2h9(0)l =1 1 (3.6)

g5 _ 1,(m_,b)

where m] = a

k6
evaluating the non-dimensional convective term using
1:200 W/m K, 11:10 W/m2 k, 5=2.1x10'6 m, and b=1.5x10°3 m
yields 0.026. This analysis revealed that 2.6% of the heat generated is lost to convection.
therefore heat lost by convection is small and can be neglected. With the effects of

radiation and convection neglected the governing differential equation can be written as

527' k 5T
k___ + —— + = O 3.7
072 r a g ( )
with boundary conditions
dT
— = 0 and TI", =Constant

dr

r=0

30
The solution to this problem is

ab2

T =T+—
(r) ” 4k

[1-(%)3] (3.8)

This is the simpliﬁed model for the radial heat conduction problem.

31

CHAPTER 4

SAMPLE DESIGN

Diamond ﬁlms are one of the most difﬁcult materials to analyze thermally. Due
to their microstructural size and fast thermal response, they pose problems in the space
and time domains (Herr, 1993). Because of these problems, optimum sample design is
extremely important in order to comply with the experimental model discussed
previously. This chapter discusses the complete fabrication of the diamond ﬁlm sample
from the wafer preparation to gold deposition, and explains why each procedure is

necessary to run the experiment and acquire accurate temperature data.

4.1 WAFER PREPARATION

The ﬁrst step in the design of the samples is to prepare the wafer for diamond
deposition. This part is crucial because if the silicon wafer is not properly cleaned and
pre-treated, diamond will either not be deposited or it might not adhere to the silicon
surface. The 4 inch diameter wafer (with silicon dioxide) is thoroughly cleaned using
pure acetone and then methanol. It is then rinsed with (distilled) water before it is dried

with nitrogen gas. This procedure is performed to verify that there are no debris or dust

32

particles on the silicon wafer. In order to ensure that no water droplets remain, the wafer
is place in an oven at an elevated temperature of 200°C for 20 minutes. After the wafer
has been properly cleaned and dried, a solution called photoresist is applied to the surface
of the Si02 (the silicon dioxide is important because it serves as insulation between the
silicon and diamond when an electric current is passed through the diamond). Since
photoresist is applied in droplets, the wafer is placed in a spin cycle for about 2 minutes
at 2300 rpm. Spinning the wafer at this rate allows the photoresist to be distributed
evenly across the surface. Finally, the wafer is placed in an oven preheated to about 400
°C for approximately 30 minutes. The heat treatment is included to ensure that the
photoresist adheres to the surface. The application of the photoresist is very important in
the deposition of diamond because it is this solution which contains diamond powder and
other elements necessary to grow diamond crystals. This completes the pre-deposition

phase of the silicon wafer.

4.2 MASK

While the mask is not needed for diamond growth, it is necessary for selective
growth. In order to grow diamond in some areas and not in others, a mask was patterned

which allowed for the exact shape and size of the diamond ﬁlms.

Masking (or photolithography) takes place in two basic steps. First, the pattern on

the mask is transferred onto the thin layer of photoresist. Photoresist is a light sensitive

33

solution similar to the coating on a regular photographic ﬁlm. It changes structure and
properties when exposed to light. Second, the transfer of the pattern takes place from the
photoresist layer into the wafer surface layer (Van Zant, 1990). Therefore, the mask
actually blocks UV rays which bombard the photoresist after it has adhered to the surface
of the wafer. By exposing the photoresist to UV radiation, diamond growth occurs in

regions shielded by the mask and is eliminated in the exposed regions (Herr, 1993).

4.3 DIAMOND DEPOSITION

After the wafer has been properly prepared and the mask generated, diamond can
now be deposited on the silicon substrate (with SiOz). This section provides a brief
description of the hot ﬁlament chemical vapor deposition system used to grow diamond
crystal on silicon substrate. Also in this section, the conditions and procedures speciﬁc
to the diamond ﬁlms used in the present work to experimentally determine thermal

conductivity are described.

The Hot Filament Chemical Vapor Deposition system(HFCVD):

The HFCVD reactor used here was a modiﬁed version of the system originally
built at Ford research lab and located in Microstructures Materials Laboratory at
Michigan State University (under the supervision of Dr. Aslam, Dept of Electrical

Engineering). A schematic diagram of the system is shown below

Mass flow
controllers

 

 

 

. v'
40*

 

 

 

Filament
power
supply

 

 

 

 

 

Pressure
controller

Boron powder
holder

f

‘— Temperature
~ / controller

Figure 4.1 Schematic of the HFCVD system (Microstructures Lab)

 

Sample holder
(4 inch water)

"I'IIIO‘

 

 

 

 

 

 

Figure 4.2 Photograph of the HFCVD system

The HF CVD reactor consists of an 18 inch diameter stainless steel chamber with a 10
inch access door. Inside the chamber are ten tantalum (Ta) ﬁlaments supported by a
square frame. The ﬁlament is design to ensure uniform deposition over a 4 inch wafer.
The ﬁlament typically draws 20-23 amps of current at temperatures of approximately
2300 °C. The ﬁlament temperature is monitored by an optical pyrometer through a 6 inch
glass port. The gas mixture inside the reacting chamber consists of ultra-pure grade
(99.995%) methane (CH4) and hydrogen (H2). The ﬂowrates of these gases are
independently controlled by MKS 250 pressure controllers. The operating pressure is
measured by a MKS type 122A baraton pressure gauge and is controlled by a MKS 250
pressure controller via a type 248A upstream valve. The base pressure is measured by a

MDC thermocouple vacuum gauge with a range of 1-1000 mtorr. A Varian SD-200

36

rotary pump is used for evacuation (to reduce chamber pressure). In addition, nitrogen is
used to purge and backﬁll the chamber. Either 4 inch wafers or graphite plates can be
used as sample holders. The supporting frame for the ﬁlament is speciﬁcally designed
with a vertical movement mechanism to adjust the separation distance between the
ﬁlament and the substrate. Two type K thermocouples are used to measure the
temperature of the top and bottom side of the substrate during the deposition process.
Since a large amount of heat is generated by the square ﬁlament array, no additional
substrate heating is required to maintain an average substrate temperature of

approximately 900 °C (Yang, 1996).

Operating procedure:

The properly cleaned and dried silicon wafer, along with the mask, is placed
inside the reacting chamber. The doping source (Boron powder) is positioned precisely
on the silicon wafer. A diagram showing the silicon wafer, which is masked for sample

conﬁguration, and boron source location is shown in Figure 4.3

37

 

 

 

/sl,licon wafer

 

 

  

 

 

 

 

 

boron
sou ce

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3 Silicon wafer with mask
The numbers correspond to each sample and were chosen for simplicity. The sample
number described later corresponds to each ﬁlm location on this ﬁgure. The silicon
substrate and the boron source are loaded on the sample holder. The distance between the
ﬁlament and the substrate is then adjusted so that the ﬁlament is only a few centimeters
above the substrate. The chamber is then sealed and evacuated to a base pressure less
than 10 mtorr. Hydrogen is then introduced into the chamber. The addition of hydrogen
causes the chamber pressure to rise. When the chamber pressure is above 20 torr, the
current to the ﬁlament is adjusted in order to raise the ﬁlament temperature to about
2300C. A slow adjustment of the current is desired to avoid damage to the ﬁlament. The
methane (CH,) is turned on after the substrate temperature reaches the desired value of
900C. With the desired settings, the chamber is left without any adjustments for the time
of deposition. Previous studies have shown that the deposition rate is approximately 0.2

micron/hour depending on the settings. After the desired deposition time has elapsed, the

38

next step is the shut down of the system. The process is terminated by ﬁrst turning off
the methane, while the hydrogen remains on for another 5 to 10 minutes. During this
period, the hydrogen is the preferential etchant to remove a thin carbonaceous layer on
the surface of the CVD diamond ﬁlms (Yang, 1996). The applied ﬁlament current is
slowly decreased to zero amps. The chamber is then returned to atmospheric pressure

and cooled. Samples can then be removed from the system.

Operating Conditions: (Typical range is speciﬁed and actual conditions used are given

in parenthesis.)

1. Gas Composition: CH4zH2=O.5% - 1% (1%)
2. Gas ﬂow rate: CH4=l-5 sccm (4sccm)
H2=200-5003ccm (400sccm)
3. Filament Temperature: 2200-2350 °C (2300 0C)
4. Substrate Temperature: 850-950 °C (900 °C)
5. Operating Pressure: 40-50 torr (50 torr)
6. Substrate to ﬁlament distance: 0.5 - 1.5 cm (1 cm)
7. Doping source: Boron powder or plane boron source or boron gas (Boron

Powder)

4.4 ETCHING

Continuing with the sample design part of the experiment, the samples must now
be shaped and etched in order to have a free standing diamond area (with no silicon
backing). Since the silicon substrate is 4 inches in diameter and a mask was used which
allowed for many separate ﬁlms on one substrate, it is necessary to separate each ﬁlm.
The cutting was done by simply scribing with carbon tip pen and then breaking the
silicon along each of the masked regions. With each sample now separated from the bulk
wafer, the resistivity and thickness measurements can be performed. The measurement of
resistivity and some of the thickness measurements pursued are described in the next
chapter (some techniques for measurement of thickness were performed before etching of

the silicon).

The silicon is removed by chemical etching using a combination of acids. The
etchant used is a modiﬁed version of the well known HNA (Hydroﬂuoric acid, Nitric
acid, and Acetic acid). The HNA etchant was chosen due to its rapid etch rate and
versatility. HNA is a very complex etch system with variable etch rate and etching
characteristics which depend on the silicon dopant concentration, the mix ratios of the
three etch components, and the degree of etchant agitation. The basic principles of

etching take place in four basic steps (Peterson, 1982). These steps are:

40

1) injection of holes into the semiconductor to raise the silicon to a higher
oxidation state Si+

2) the attachment of hydroxyl groups OH' to the positive charge Si+

3) the reaction of the hydrated silicon with the complexing agent in the solution

4) the dissolution of the reacted products into the etchant solution.

The four steps imply that any etching solution must provide a source of holes as well as
hydroxyl groups, and must also contain a complexing agent whose reacted species is
soluble in the etchant solution. In the HNA system, both the hole and the hydroxyl
groups are effectively supplied by the strong oxidizing agent (HNO3), while the ﬂuorine

from the HF forms the soluble species (Peterson, 1982).

Based on the chemistry described above, the modiﬁed version used consisted of
Nitric acid and Hydroﬂuoric acid in a 3 to 1 ratio. This combination was primarily used
in the initial phase of the etching process, while 2:1 combination of nitric and
hydroﬂuoric acids were used as a second step when it became apparent that the diamond

was exposed.

The etching procedure itself is quite simple, however it is rather time consuming
and must be performed meticulously. The solution is applied to the silicon side in small
droplets using a micro pipette. It’s important that the solution is applied to the same

region each time, where it attacks the exposed silicon. Since the solution is applied to the

41

silicon in tiny amounts, it acts as a limiting reagent. Therefore, after a short time the
reacted mixture is carefully removed and small droplets of the solution are reapplied.
The process is repeated for several hours (IO-16) until the diamond is exposed. The
etching process is only terminated after a 3mm diameter of free standing diamond is
produced. The entire process is tedious, and while the solution is being added and
removed numerous times, each movement must be done very carefully in order not to
damage the even more delicate sample. Figures 4.4 and 4.5 show the sample after the
etching is completed. The samples are approximately 2.0 cm x 2.0 cm with the diamond
ﬁlm occupying 1.5 cm x 1.5 cm. The etched region is about 3mm in diameter. The

etched angle vary depending on how the solution was applied.

Etched

Silicon \ ”91°" / Stllcon
L ﬁlm I

 

 

 

 

 

 

 

Figure 4.4 Side view (cut away) of the etched diamond sample

 

. ' 1

I- a J\ -" .
51" l /\r\"‘. ‘

I‘QQICJ".

--.~.... .._.___..... "A

Figure 4.5 Top view of the etched diamond sample

42

4.5 SAMPLE PREPARATION

With the completion of the etching, it becomes necessary to produce the
temperature rise within the diamond region. This is done by creating an electric circuit
within the diamond. Two strips of electrically conducting tape (manufactured by
Electrolock Inc in Chargrin Falls, OH) are fabricated for placement on the diamond
surface. Because of the miniature size of the diamond sample, the width of the
conducting tape was made smaller at one end. A diagram showing the shape of the

conducting strip is given in Figure 4.6.

 

 

/

Figure 4.6 Conducting strip

The conducting tape, with one adhesive surface, is then attached to the surface of the
diamond. Lastly, to ensure good electrical contact between the tape and the diamond,
electrically conducting silver paint manufactured by Dynaloy Corporation (“340”) is
added to the diamond-tape interface. Figure 4.7 below illustrates the sample setup and

Figures 4.8 and 4.9 show photographs of the sample from both sides.

43

 

 

 

 

dictum!
Etch region
silicon
c uctive tape
Figure 4.7 Sample setup

 

Figure 4.8 Photograph of sample viewed from diamond side

44

 

Figure 4.9 Photograph of diamond sample viewed from the silicon side

4.6 GOLD DEPOSITION

Due to the spectral characteristics of diamond and the use of infrared
thermography to acquire temperatures, one additional requirement must be satisﬁed in the
sample design. In order for infrared thermography to yield accurate temperatures, the
sample must be opaque in the infrared and the emissivity must be known. The
determination of the emissivity of CVD diamond is explained in the next chapter. In this

section, the method used to ensure that an opaque sample is achieved is discussed.

45

The sample is made opaque by overlaying a thin layer of gold on one side of the
diamond. The added layer serves two purposes, 1) it reduces the sample electrical
resistance (not the diamond), therefore allowing for electrical resistance heating with a
relatively low voltage input, and 2) it creates a sample which is opaque in the infrared,
therefore allowing for accurate temperature readings with the use of an infrared
thennograph system. Without an opaque sample, the scanner would be measuring
temperatures which are a combination of the diamond temperature and the background
temperature. In lowering each sample’s electrical resistance, the coated layer allows
heating for all levels of boron d0ped diamond. Without such a feature, only a small range
of doping level could be investigated for thermal conductivity using the techniques
discussed in this report because at lower doping levels the electrical resistance would be

too high.

The procedure for precise sputtering of a thin gold layer is simple. The Emscope
SC500 Sputter Coater (located in the Center for Electronic Optics at Michigan State
University under the supervision of Dr. Flegler) is used with a gold target (photograph

shown in Figure 4.10).

46

 

Figure 4.10 Photograph of the Emscope SC 500 sputter coater

A mask is used to ensure gold deposition on the diamond, and not on the silicon
substrate. It is imperative that the gold generate energy in the diamond and not in the
silicon, otherwise the heat sink characteristics of the silicon wafer necessary for the
experimental model will not occur. Since the gold is deposited using a sputter coater, the
mask simply involves any material which shields the silicon wafer from falling atoms.
The masked sample is placed in the sputter coater chamber where a vacuum is created.
The instrument is allowed to coat at 20 rnA and 0.1 torr. At this setting, the Emscope
SC500 coats at 70 angstroms per minute. The instrument was allowed to coat for 14
minutes producing a predicted total thickness of 980 angstroms. This thickness was not

veriﬁed experimentally. Because the coated layer is only a small fraction (less than 4%)

47

of the diamond’s thickness, it’s effect on the measured thermal conductivity is assumed

to be small.

48

CHAPTER 5

EXPERIMENTAL METHODS

The previous chapters describe the sample setup and the experimental model. In
this chapter, the equipment and method used to acquire temperatures and estimate the
thermal conductivity are discussed. However, before experiments can be performed,
thickness and resistance measurements must be made. Also, it is necessary to understand
the basic concepts needed for accurate temperature measurements using an infrared
thermography system (this includes knowing the emissivity of diamond). Once accurate
temperatures are determined experimentally, the methods of parameter estimation are

used to determine the thermal conductivity.

5.1 EXPERIMENTAL SETUP

Due to the extensive requirements involved in the design of the samples, very few
items are needed for the experimental setup. The setup consists of:

1) Power supply

2) Digital multimeter

3) Diamond sample

4) Adjustable clamp

49

5) Electrical lead wires

6) Infrared camera and control unit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7) Jiffy Jack
war Supplu
OED
Dtmond
ear-ole .
Scanner n L
”If." m.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Image Processor

FIGURE 5.1 Schematic diagram of experimental setup

 

Figure 5.2 Photograph of data acquisition system

The setup, shown schematically in Figure 5.1 and photographed in Figure 5.2, consists of
two main regions. One region serves as the heat generation area within the diamond, and
the other region serves as the temperature sensing area. The energy generated within the
diamond is accomplished through the use of an electrical circuit formed with the
diamond. This region consists of the diamond sample, the electrical lead wires, and the
power supply. Figure 5.1 (right hand side) shows the necessary connection required for
energy generation in the diamond. The temperature sensing area consists of the infrared
camera (scanner) and control unit. The scanner is focused on the diamond window where
internal energy generation takes place, while appropriate settings to acquire accurate
temperatures are selected on the control unit. The remaining items are used to aid in the

setup and data acquisition.

51

5.2 INFRARED THERMOGRAPHY

Because of the miniature size and fragility diamond ﬁlms, conventional methods
such as the thermocouple and the resistance thermometer are unreliable to acquire
temperature. As a result, an infrared imaging camera was used to image microstructural
areas. In this section, the basic components of an infrared system and all equipment used
to acquire temperatures, along with parameter settings used while performing the

experiments are discussed.

The system used is the model 600L imaging radiometer (Figure 5.3). This system
is designed for applications requiring accurate real time analysis of static or dynamic
thermal patterns. The high performance system combines superior image quality and
thermal sensitivity with true temperature measurement display, digital remote control,
and automatic emittance and background correction. The system can be combined with a
video cassette recorder (VCR) for later analysis on playback. While operating, the
thermal reference target is viewed by the detector 60 times per second. A
mercury/cadrnium/telluride detector, while being cooled by liquid nitrogen to 77 K within
an evacuated metal cryogenic dewar, is used to produce maximum thermal sensitivity and

high spatial resolution (Inframetrics Operator’s Manual, 1988).

52

 

Figure 5.3 Photograph of the Model 600L (Scanner and Control Unit)

The Model 600L scanner incorporates electromechanical servos (galvanometers)
to perform horizontal and vertical seaming. This scan concept produces the highest scan
speed or frame update rate of any imaging radiometer, permitting fast moving targets to
be viewed without signiﬁcant distortion. The optical path of the scanner is shown in

Figure 5.4.

53

 

  

HORIZONTAL
— SCAN MIRROR
m

VERTICAL
SCAN MIRROR
corn

 

 

 

 

 

Figure 5.4 Optical Path of the Scanner (Inframetric Operator’s Manual, 1988)

Thermal radiation enters the evacuated scan module through a germanium window, is
deﬂected by horizontal and vertical scan galvanometers, and exits through a second
window to pass through the focus lens on the detector. During system operation, a
magnetic ﬁeld oscillates a mirror against a torsional spring shaft under closed-loop
electrical control. Horizontal scanning is performed at a very high rate (4 Khz) in a
resonant (sinusoidal) mode. Vertical scanning is done in a sawtooth pattern similar to

standard TV formats (Inframetrics Operator’s Manual, 1988).

54

The model 600 control unit contains circuits to process, digitize, reformat, and otherwise
prepare the IR signal for display. A microprocessor performs internal calibration as the
scanner temperature changes, ﬁlters are installed, or settings are altered. The
microprocessor accesses individual picture elements (pixels), then calculates temperatures
using internal calibration tables corresponding to the optical ﬁlters/lens combination in
use. A pulse-width modulated switching power supply accepts a wide range of DC input

voltages with excellent noise immunity (Inframetrics Operator’s Manual, 1988).

Two additional lenses are used to improve the spatial resolution necessary for the
3mm diamond area. The afocal 3X telescope uses four germanium optical elements
which reduces the system ﬁeld of view (FOV) and resolution element size by a factor of
3. The bayonet-type mounting is equipped with Hall-effect sensors which reinvert the
image for proper display. Attached to the 3X lens is a 6 inch closeup lens which can
resolve objects smaller than 100microns (Herr, 1993). Photograph below, Figure 5.4,
shows each lens used. The lens on the left is the 3X telescopic lens. The lens on the right

is the 6 inch closeup lens.

   

Figure 5.5 Photograph of Lens

5.3 DETERMINATION OF EMISSIVITY

One of the difﬁculties encountered when using the infrared thermography system
to measure temperature is knowing the emissivity of the object being measured. In this
section, the importance of knowing the emissivity is discussed, along with the
experimental setup, techniques, and results used to determine the emissivity of CVD

diamond.

In order to convert the emissive power measured by the radiometer to surface

temperatures, the emissivity of the diamond window needs to be pre-determined. As

56

described in the sample design, the diamond window consists of CVD diamond and a
very thin layer of gold. Therefore, the emissivity of this combination (diamond and gold)
is required. Without an accurate measurement of the emissivity, temperature

measurements from the radiometer will be erroneous.

The experimental setup consists of a black aluminum plate heated with a constant
heat ﬂux on the backside. A thin (2mm) aluminum plate is sprayed uniformly with black
paint (Champion sprayon high heat 1200 °F) on both sides. A ﬂat plate heater is then
attached to the aluminum plate using conductive thermal paste (Omegatherrn “201”).

Figure 5.6 illustrates the black plate with attached heater.

Thermal Paste To

     
  

' ‘ ' 6.6.333???8’8’8’8’8’8’34'.

 

 

Figure 5.6 Black plate with heater attached

A thermocouple is then attached to the front side of the plate using an epoxy (Devcon 5
minute epoxy). The sample is then suspended in front of the scanner to check the

isothermal nature of the plate, and subsequently used to determine the emissivity.

57

The ﬁrst step is to check the isothermal nature of the black plate across a region of
interest, and then verify that a steady state condition has been reached. The latter is
checked by simply applying a voltage and then monitoring the temperature changes for a
period of time. While several voltage settings were tested, a power supply reading of 7
volts worked best. With this applied voltage, a steady state temperature was achieved
quickly, and the temperature was monitored for 20-25 minutes. Steady state temperatures
remained ﬁxed (within the radiometer’s uncertainty of 0.1 0C) if not affected by forced
convection. Therefore, in order to avoid sudden changes due to air circulation in the
laboratory, the sample was properly shielded by placing cardboard around the sample.
Constant temperature across the plate was investigated by focusing a region of the black
plate on the RGB monitor. While using a temperature span of 10 degrees, the electronic
ruler of the ThermaGram software program was used to show the temperature distribution
across the region. This method was ﬁrst used, however in order to verify what is seen on
the monitor, temperatures were extracted. Because of the noise level involved in using
infrared thermography, data for the black plate at room temperature were ﬁrst extracted.
The temperatures were measured over a length of about 1.5 inches along the black plate.
A plot of these data is given in Figure 5.7. The room temperature data is important
because it shows the best possible accuracy to which data can be obtained (average=24.01

i 0.08).

58

2438 . 2 .. _. .. _ . .......
24.2I

24151

 

 

 

 

 

 

 

 

 

 

 

0 50 too 150 zoo 250 eon
“Hum

Figure 5.7 Black plate at room temperature

The heated black plate at steady state is then extracted, and shown in Figure 5.8.

 

 

31.95 .

 

31.. {r

 

 

 

Yw ICI
L
*—-—-;—"
.__a
ﬂ;
:2
at;
.0—
a

 

 

 

 

 

 

31.” .
~ I
31.8 I I
«I 1 4
31.75 I-.-.__. .. -H..i _. _._-. . ---.._ﬁ_ .-_- n h-.- _. i-.. ,--. . .._. - i
0 so roe 1” an no son
Willa”!!!

Figure 5.8 Temperature distribution of heater black plate

59

Observation of temperature data for Figures 5.7 and 5.8 shows a similar peak to peak
noise level of i025 °C. Figure 5.8 shows that the heated plate is isothermal with a

measurement uncertainty of 0.25 °C. This corresponds to a mean of 31.93 and a standard

deviation of 0.07.

A diamond sample is subsequently etched and coated with gold as described
previously. The sample is then attached to the plate using a high conductive thermal
paste. Figure 5.9 below shows the diamond sample attached to the black plate (with
heater on the backside), along with a type B thermocouple. The black plate is 3 inches in

diameter. The thermocouple was located about 0.5 cm from the sample.

 

memrhordiedremon

Figure 5.9 Black plate with attached sample

With the black plate suspended in front of the IR scanner, the point mode function (used
to measure temperature at a given location on the object) on the radiometer control unit is

used to measure the emissivity of both the black paint and the diamond/ gold region

60

(which faced the scanner) of the sample. The premise used to ﬁnd the emissivity is to
match the temperature measured by the radiometer to the “true” temperature of the body
(as measured by the uncalibrated type B thermocouple). The ﬁrst step is to ﬁnd the
emissivity of the black paint. This is accomplished by adjusting the emissivity setting on
the control unit until the temperature of the scanner at a point close (<05 m) to the
thermocouple matches the temperature measured by the thermocouple. Based on the

experimental data in Table 5.1, the emissivity of the black paint was taken to be 0.98.

 

Table 5.1 Emissivity of black paint

With this value of emissivity for the black paint input into the control unit, the correct
temperature of the black plate can be read from the RGB monitor using the point mode
function. Still using the point mode function, the emissivity of the diamond window for
the attached sample is determined. The procedure is similar to that used for the black
paint, however the reference temperature (true temperature) is that of the black paint with
an emissivity of 0.98. The emissivity of the diamond is then adjusted until the
temperature of the diamond matches that of the black plate (e=0.98). Tables 5.2 to 5.5

below show that the temperatures match closely at emissivity values (of the

61

diamond/gold region) in the range of 0.61 to 0.63. Due to noise level in the system, four

experiments were performed and an average emissivity was found to be 0.62.

 

Table 5.2 Emissivity of diamond (test #1)

62

 

Table 5.3 Emissivity of diamond (test #2)

NWT“ CV5 "I" "X PETE." . -

m ’-- A.‘ J "' $13.28.. and». .AB'JLA-L'A‘ u 15: ~. 3...

0.97 41 '

 

Table 5.4 Emissivity of diamond (test #3)

63

 

Table 5.5 Emissivity of diamond (test #4)

The determination of the emissivity of diamond/gold window using the above technique
is advantageous because it measures temperatures from an infrared system similar to
those that would have been obtained had a type E thermocouple been placed on the
diamond window. The emissivity of 0.62 compares well with Herr’s measurement of
0.63. Herr’s (1993) experimental setup measured the diamond with the silicon backing
present, while this work measured the diamond/ gold combination. This would indicate

that the radiometer is only detecting emitted energy from the surface of an opaque object.

64

5.4 MEASUREMENT OF RESISTIVITY

While there are different techniques to measure the resistance of diamond, the
four probe method was chosen due to it ease in measuring resistance at local areas on
each sample. A brief description which explains how the method is applied and how

resistivity is calculated is given in this section.

In the four probe method (shown in Figures 5.10 and 5.11), four equally spaced
point probes are brought into contact with the diamond surface (before any preparation
conditions are performed), and a known DC current is passed through the outer probes.

The two inner probes are connected to a high impedance voltmeter.

AIII'IOI‘OI"

 

 

7’2"

 

 

'——=—1|II

 

 

 

Figure 5.10 The Four Probe method

65

 

Figure 5.11 Photograph of the Four Probe instrument

This arrangement largely eliminates contact resistance effects, since the voltage-
measurement probes draw negligible current (Reinhard, 1987). The relationship between

sheet resistivity, RD (ohms), voltage, V(volts), and current, I (amps), is given by

(5.1)

where C is a constant which depends on the ratio d/s and a/d (d is the sample length, a is

the width, and s is the space between the probes). Based on these ratios, values of the

66

constant C are extrapolated from Table 7.1 (p.346) of Introduction to Integrated Circuit

Engineering by Reinhard. The resistivity, p (Q cm), is calculated using

,0 : 6RD (52)

where 5=average sample thickness and RD=sheet resistivity.

The procedure is quite simple. Each sample is placed individually across the four
probes, and a known voltage is applied (about 10‘6 amps). The voltage across the inner
probes is subsequently recorded, and used to calculate resistivities. Values measured for

all samples used are given in Table 5.6.

 

 

Esme . vaaeﬁlaﬁvllﬁaw Ruisﬂvltx.
Number. (ohm Corn)~ I ’(ohrn cm)
_. __1L. -- ._-._ _ I"? 1-53
_ “191”.” F 3.98 - 2.2g
‘_ _ -92.- 077... f 5.61
_ __g _7 8.07 6.75
_’ 6': ” E “11.68“" " ‘ "€82“
_ 3. ’ 18.40‘ ' ‘ ’ 14.96
_ 4; 28.53: 21.85:
_ _ 2.._ 31.68 28.38
1 ‘ 80.14 ' 68.45

 

 

 

Table 5.6 Resistivity of samples

While the four probe method measures local resistivity, its possible to calculate global

resistivity by measuring the resistance for a known length and subsequently calculating

67

the resistivity. The DMM probes were placed in contact with the electrical conducting
tape. The measured resistance was then recorded along with the samples width, length,
and thickness. These values were then used to calculate global resistivty (p=RL/A) This

was done for sample #2 and #3, and the global resistivities are shown in Table 5.7 below.

 

 

 

Sample Number Resistivity
(Q cm)
2 20
3 1 6. 8

 

 

 

 

Table 5.7 Global resistivity using a DMM

Comparing the global resistivity to the local and average from Table 5.6, it can be seen
that for sample #2 the global is somewhat lower, however for sample #3 the global value
is between the average and local values. Because only two samples were used and the
variability of doping when local measurements were made, little information can be

deduced from the comparison.

5.5 MEASUREMENT OF FILM THICKNESS

The measurement of diamond ﬁlm thickness was necessary because accurate
estimate of thermal conductivity depends on accurate measurements of all parameters.
This proved to be challenging since the ﬁlms were on the order of a few microns. Several
methods were pursued in order to accurately determine the thickness and thickness proﬁle
of each diamond sample. In this section, a brief description of each method is given,

along with results and explanations of the accuracy of each method.

68

The ﬁrst instrument tried was the Sloan Dektak IIA proﬁlometer which is located
in the Physics Department at Michigan State University (under the supervision of Dr.
Golding). This instrument is basically a surface roughness measuring device. The basic
theory behind this instrument involves detecting the step change between two surfaces. in
this case diamond and silicon. Figure 5.12 illustrates how the Dektak actually measures

thickness.

Stulua

 

l Dtgyond 1
Silicon

 

 

 

 

Figure 5.12 Sloan Dektak IIA

Since the diamond was deposited on the silicon using a mask, the step change which
occurs at the edge of the diamond-silicon interface can hopefully be measured. While the
idea is rather simple, the application presented uncertainty in the uniformity of the
diarnond’s thickness. One source of problems encountered when using the instrument
was the leveling of the reference surface. Since the silicon surface assumes the slope of
the surface on which it sits, a small slope in this surface will cause a sloping of the
silicon, and consequently the diamond. This unevenness will be detected and affect the

results. Therefore it is difﬁcult to distinguish whether the thickness is uniform. Another

69

problem encountered was in the use of the silicon as reference. Because of the
fabrication techniques used to separate each sample from the bulk wafer and the heat
treatment necessary to grow diamond, the silicon possesses a spherical shape, and could
not serve as a reliable reference. These are just some of the problems encountered in

using the Dektak to measure thickness.

Fairly accurate results can be obtained if it is assumed that the thickness of the
diamond is uniform over small regions (2 mm). Based on this assumption, the leveling
feature can be used to distinguish the diamond surface height from the silicon reference.
The step change can then be measured using calibrated scales within the instrument.

Results from the Dektak are given later.

Another instrument used with a great deal of success is the WYKO RST-PLUS
rough surface/step tester. The tests were performed at Wyko Corporation in Tucson, AZ
by Tom Stoudt. The RST-Plus is a non contact proﬁler capable of measuring step height
from 100 angstroms to 500 microns. Along with thickness, the RST—Plus is capable of

detecting the thickness proﬁle of both the silicon and the diamond.

The technique used by the RST-Plus for determining surface height using phase-
shifting interferometry typically involves sequential shifting of the phase of one beam of
the interferometer relative to the other beam by a known amount, and measuring the

resulting interference pattern irradiance. The phase data are then processed by an

70

algorithm that removes phase ambiguities between adjacent pixels. The surface height
can then be determined from the phase data. One disadvantage of this technique is its
limitation to smooth continuous surfaces, because reliable removal of discontinuities
depends on the phase not changing by more than 1'1 (Caber, 1996). Measurement of step

heights for two samples (#17 and #3) are shown in Figure 5.13 and 5.14 below.

 

Figure 5.13 Step heights using RST-Plus (Sample #17)

 

Figure 5.14 Step heights using RST-Plus (Sample #3)

The measurements were obtained using a 20X objective. The 20X objective permits
sampling a 228nm by 304nm area with a spatial sampling interval of 0.8 microns. The

thicknesses obtained are shown in Table 5.8

71

 

Table 5.8 Thickness measurements using RST-Plus

To measure the thickness proﬁle, an alternative method of determining surface height
using white light as the source in an interferometer is used. The method measures the
degree of ﬁinge modulation, or coherence, instead of the phase of the interference fringes.
The principle behind white light vertical-scanning interferometry utilizes the idea that
when white light is used as a source in an interference microscope, the modulation, or
visibility of the fringes, drops off rapidly from its maximum value at minimum OPD
(Optical Path Difference). If the modulation signal is extracted from the intensity signal
as the OPD is varied through focus and its peak is detected, a measurement of relative
surface height at that point can be obtained (Caber, 1996). If this procedure is done for
each point, a three dimensional surface map can be obtained. The 3-D plot represents the

thickness proﬁle of the surface.

In order to verify the proﬁle over the entire surface, measurements were taken at
the comer of each sample (Figure 5.15 and 5.16) as well, and this conﬁrmed that the
shape of the diamond was due to the silicon substrate. The fringe patterns shown in
Figures 5.15, 5.16, and 5.17 is due to surface heights at each point on the sample. The
curvature indicates that constant height occurs at regions which form a circle (similar to a

cone).

72

\
\

I

74:4 . --‘ 1;;
_ ;.-.;.;-:..; : .4 .
‘1.,\'-.\:‘\‘-\>\.‘\§::::;:.;:m ' .. .
\ GSW 3
‘- '- \fi-I‘~\\. "s. «11“. '- .
§ I I I‘I:'\':"}‘:‘.‘~T:::T:.‘Zi,; '
N ._ ,
‘ _
\\\\\\\\\\\‘v

 

Figure 5.15 Corner proﬁle of thickness (Sample #17)

 

Figure 5.16 Comer proﬁle of thickness (Sample #3)

73

 

Figure 5.17 Bullseye at the center for spherical shape (Sample #3)

The interferogram at the center of the silicon (sample backside) for samples 1 shows a
circular “bullseye” fringe pattern which indicates a spherical surface (Figure 5.17),
convex in this case. The Figures shown above were measured with a 1.5X magniﬁcation
objective and a 0.5X converter. This combination is equivalent to a 0.75X objective that
proﬁles a 5.5 mm by 7.4 mm area with a spatial sampling interval of 20 microns. The
results clearly show a convex shape (Figure 5.17) of the diamond ﬁlm and silicon

substrate.

A Laser Scanning Confocal Microscope (LSM), located in the Laser Lab in the
Crop and Soil Sciences Dept. at Michigan State University under the supervision of Dr.

Whallon, was another instrument which was used in order to accurately measure the

74

thickness or step change of the diamond. The words “laser” and “scanning” refer to the
method of illumination, while “confocal” refers to the method of image formation (not all
confocal microscopes use laser or scan the sample). A confocal microscope is aligned so
that the illuminated spot and the image spot coincide precisely. This alignment is not
present in all microscopes, therefore light reaches the observer frames from all parts of
the sample within the ﬁeld of view. Light from out of focus areas severely degrades the
focal plane image, thereby limiting the depth to which a sample can be examined. In a
confocal microscope, the sample is scanned point-to-point with a ﬁnely focused laser
beam, and a pinhole aperture is placed directly in front of the detector at the focal point of
light coming from the in-focus part of the sample. The effect of these modiﬁcations is to
block light from out-of-focus regions because the focal point of such light falls either in

front of or behind the pinhole aperture (Whallon, 1996), as shown in Figure 5.18.

/Beam Splitter
Laser

Collecting Lens

IAIEI

/Objective Lens Confocal Unit
With Pinhole
Focal Plane

/Out 01- Focus Area

math of In-Focus Light

 

Figure 5.18 Focal plane of LSM

75

The result of this instrument conﬁguration on the ﬁnal image is a dramatic increase in

resolution and contrast.

One particular feature of the LSM which allows for microscopic thickness
measurements is a feature known as optical sectioning. In this process, the microscope
collects a series of optical sections at intervals along the z—axis. The application of a
feature known as phi-Z section produces an image of the sample as it would appear if cut
in the vertical plane. This image is then ﬁt to calibrated length scales within the system,
where the distance from the top of the diamond to the top of the silicon is measured.

Figures 5.19 and 5.20 show thickness measurement along with confocal images of the

diamond surface.

     

‘ ". I ‘30 I - I? 1‘ .

x s, e _ . . . e ‘ . .- _ _‘

Figure 5.19 Thickness measurement on the left edge of Sample #3 (8=3.1 um) using
LSM

76

2008548

 

Figure 5.20 Thickness measurement on the right edge of sample #3 (8=3.0 um) using
LSM

One disadvantage of using the LSM was the uncertainty involved. The 100X objective

used had an uncertainty of about 1 microns. This poses a problem since the samples

range in thickness from 1.5 to 3 microns, therefore the relative uncertainty ranges from

33% to 66% making the measurements meaningless. Also, the LSM was not successful in

detecting the thickness proﬁle of the diamond and/or the silicon because of the limited

ﬁeld of view (approximately 0.5 mm) capability of the lOOX objective.

77

The NEWVIEW 100 by Zygo Corporation in Middleﬁeld, CT(tests performed by
John Roth) was used to analyze the shape proﬁle and determine ﬁlm thickness. The
overall results were mixed. The Newview100 was successful in determining the
thickness proﬁle, however its measurements of ﬁlm thickness exceeded values measured
by other instruments. The measurements of ﬁlm thicknesses were approximately 6
microns which is 2 to 3 times that of the RST-Plus (1.5-211m), LSM (~3 um), and Sloan

Dektak (2pm).

The Newview 100 is a scanning white light interferometer. It works by using
traditional techniques in which a pattern of bright and dark lines (fringes) result from an
optical path difference between a reference beam and a sample beam. A simple
mechanism is used to split incoming light inside an interferometer, one beam going to the
internal reference surface and the other to the sample. After reﬂection, the beam
recombines inside the interferometer, undergoing constructive and destructive
interference and producing a light and dark fringe pattern. In the system, a translation
stage and a CCD camera work together to generate a 3-D interferogram of the object.
This interferogram is stored in the computer’s memory and then transformed into a

quantitative 3-D image of sample thickness over a speciﬁed area (Zygo, 1996).

Analyses of two samples (#4 and #5) were performed using the Newview 100.
The thickness data are shown in Figures 5.21 and 5.22. Oblique plots are also given in

Figures 5.23 and 5.24.

78

 

 

 

 

 

8 299° Surface Profile
+9.m-‘
: . _‘. fv‘l‘b‘v' L)”
+5.5w38— 4“” I If)
E :
3 .1
~a+aouwo—
u -1
5 :
Emmanue—
m . _____________ ___ ._
I :
-e.aunai
r1 I I I I I I I I I I I I I I I I T I I I I I I I I I I
man 1850 1.!3 1Jn 21B 850

 

 

Distance (mm)

 

Figure 5.21 Thickness measurement on Sample #4 using Newview 100

 

 

 

 

I 235° Surface Profile 'K
ammo-1
ammol I, - III VLF VIN!
E .
D 4
V 4.133%8— UV
9 I
c .
9 2.88888-
0 I
I ‘ L..—
a.meee- --------------------------------
ﬁI I I I I I I I I I I I I I l I I I I I I I I I I I I I
B 500 1m 15% 2m 25%
Distance (um)

 

 

Figure 5.22 Thickness measurement on Sample #5 using Newview 100

 

79

I ‘85" Ublique Plot '\

+1.5627B

mm

 

 

 

 

Figure 5.23 Oblique plot of surface for sample #5

Oblique Plot

 

:- +1.53953

: . . __
‘3?! v- ‘ -
-.-_ 13.2,. UH".
. . O“, :_
“'8 1‘.‘ -.2
‘ “-22;
J ‘y‘k..:)_>)“\'." ‘ .
I"... i“h“;~"4 1‘.‘ -4 ZSEBS
- -:I" «F. , '1‘": yr -' o
125%: ~. has}. _,
‘ ' 5'13. - "ﬁb‘f: ' D l
.‘:-‘." fl.“ - < L - I -
‘Jl'é‘ﬂt.‘_t ‘. 3 I
- ;.t,_2§§3%§' Il
‘. A352“. v V " K;
.‘.- 4‘ ”I Y ‘ '

41.5." a ‘ .

l 7

.. j , _.~ .'-'.-'.-‘.' 4‘3 [I' {I

-:.-> raw-cw l“.

3; mm

 

 

9.1 rum 3.1

 

 

Figure 5.24 Oblique plot of surface sample #4

80

The measurements show a step change from the silicon surface to diamond of about 6pm

for each sample. These values are 2 to 3 times larger than other instruments used. The

surface proﬁle of each sample is shown in Figures 5.25 and 5.26.

 

 

 

 

 

 

 

 

 

 

I 189° Surface Profile 4
«imam
_. "" -- .. ' ‘-
Anxmwnj , ‘
e '* r " ~
3 1 _ \
“+2.08%? /'
a .‘ ‘
z 1 ,
onjmmna-w -----------------------------------
q; 4
I 1
~21mmn—
I I I l I . I 7 r— T
we so lee
Distance (mm)
Figure 5.25 Surface proﬁle of sample #4
I 139° Surface Profile 4
«Lemma?
1 ",-«*"“”-‘r-c
“macaw-fl. ﬂ, ‘ ‘ ~_f
E : .-'- i L . *x-‘
3 _ ;
waaaaaa -»« xx.
w _ f x
c z
cum —” -----------------------------------
%
-—” .08888
I I I T I I I 1 ﬂ I I
0.0 5.8 10.0
Distance (mm)

 

 

Figure 5.26 Surface proﬁle of sample #5

8]

While these proﬁles indicate a parabolic shape for the diamond ﬁlm, it doesn’t
distinguish whether the shape is due to the silicon or the diamond. The results from the
Newview 100 indicate that it is an instrument with a great deal of promise. However,
because of its high values for thickness and uncertainty in showing whether the shape is

due to the silicon or the diamond, further analysis was discontinued.

An Environmental Scanning Electronic Microscope (ESEM) was another
instrument used to measure thickness. The ESEM has the advantage of showing the
various layers comprising the sample. Therefore, it is easy to measure all layers (Silicon,
Sioz, Diamond) of the sample using calibrated scales within the microscope. The results

from the ESEM for samples # 5 and #14 are shown in Table 5.6.

82

 

 

 

 

 

 

 

 

Test Number Sample #5 Sample #14

1 2.0 1.48

2 1.78 1.6

3 1.85 1.7

4 1.74

5 1.7

Average 1.814 1.593

Standard Deviation 0.119 0.110

 

 

 

 

 

Table 5.9 Thickness measurements using ESEM
One disadvantage of the ESEM has to do with the preparation conditions necessary for
accurate measurements. The ESEM needs to examine the samples on its edge so all
layers can be viewed. In order to view the samples on edge, they must be broken along a
region. This limitation makes the technique destructive, therefore samples measured by
the ESEM cannot be used to determine thermal conductivity. For this reason, thickness

measurements using the ESEM was discontinued.

in summary, measurement of ﬁlm thicknesses were investigated using four
separate instruments. Each instrument was used to measure ﬁlm thickness and determine
uniformity over the ﬁlm surface. The RST-Plus by Wyko Corporation gave reasonable
results for both thickness and thickness proﬁle over the surface. However, the values for
ﬁlm thickness were lower than all other instrument. The Newview 100 by Zyko
corporation also displayed excellent capabilities in generating 3-D thickness proﬁle and
thickness measurements. However, measurements of thicknesses were higher than other

instruments, and the parabolic thickness proﬁle did not distinguish whether the shape is

83

due to the silicon or the diamond. The LSM measured thicknesses of about 3 microns,
however due to its large error (1 micron) and the sensitivity of thermal conductivity to
ﬁlm thickness (described in uncertainty analysis) the results were not used. The Sloan
Dektak IIA presented difﬁculty due to the spherical shape of the sample (as shown by
Wyko RST—Plus). Because it was understood that the parabolic shape of the sample was
due to the silicon substrate, the leveling feature can be successfully used to detect step
change in height. This procedure was performed and thicknesses (step change) were
measured. Figures showing some of the measurements performed, along with Table 5.6
of other tests are shown below. Based on the measurements from the Dektak, it was
assumed that all samples were 2 pm thick. The “spikes” in the Dektak measurements
could be due to dusk particles on the surface. The table of thickness ( Table 5.6) shows
measurements performed on sample #8 (test #2,6,8), sample #4 (test# 1,4, 5), and sample

#3 (test # 3, 7,9).

 

 

 

 

 

MM!
mm
100000 ,. - .c
N
00000 0
f D
1
0
70000
00000
f
30000 0
20000
10000
0 l
0 200 000 000 000 1000 1200 1400 1000 1000 2000
~00.de

Figure 5.27 Thickness measurement (test#3)

WWI

W (W1

84

Sloan 00111011 ll
Thickness "1000010111011:

1.

.§§§§§§§§§
+
1:

._.___..._._.._.~

 

 

A

 

 

200 400 000 000 1000 1200 1400 1000 1000 2000
Wu“. 0000-

Figure 5.28 Thickness measurement (test #7)

 

 

 

   

 

 

810.: 00111011!
11110101000 «1000100111011:
00000 __.
70000 .. j,
U :
m 4 1 l
» E
0 f
m )0 I
>1 l
l ,
4WD 7 +M81
I
00000 it l
20000
10000 . ,
0 c
0 200 400 000 000 1000 1200 1400 1000 1000 2000

Were-1000011:

Figure 5.29 Thickness measurement (test #9)

85

 

Sample Number Average Thickness (um) Standard Deviation (pm)

 

 

 

 

 

 

 

3 2.38 0.1
4 2.3] 0.37
8 1.98 0.07

 

Table 5.10 Thickness measurements using Sloan Dektak IIA

Lastly, all measurements of thicknesses are compiled in Table 5.11. One factor
which might affect the measured results for the Dektak was the condition of the sample
when measurements were made. Sample #3 and #4 were performed after various
preparation condition necessary to estimate the thermal conductivity were applied and
then removed. This condition could lead to higher values for thickness since residue

from the paint, gold, and chemicals could affect the results.

Sample RST—Plus DekTak IIA LSM ESEM NewView 100
Number ) ) ) ) )
3 1 .34 2.4 3 .0
2.3 6.0
1 .81 6.0

2.0

1.59
1.74

 

Table 5.11 Summary of thickness measurements

All other measurements were performed before any preparation condition were applied
to the samples. In order to compare the results of all the instruments used, the
measurements must be compared with consideration to measured standard deviation and
instrument precision. Table 5.12 shows each instrument measured standard deviation and
uncertainty.

86

 

 

 

 

 

 

Measured Standard Uncertainty
Deviation (um)
Dektak IIA 0.233 0.01
RST-Plus 0.283 0.05
NewView 100 0 0.5
ESEM 0. l 56 0. l
LSM 0.1 l .0

 

 

 

 

 

Table 5.12 Uncertainty and standard deviation of each instrument

Table 5.12 shows that when standard deviation and uncertainty are considered, the
samples are within the range of the measured values predicted by the Sloan Dektak IIA
except for the newview 100 which is high due to the belief that it measured both the
diamond and the silicon dioxide (about 3 pm). Also important in comparing the various
instruments is accessibility and cost. The Dektak is easily accessible and free of charge.
The LSM is also accessible (on the Michigan State University campus), however cost $34
per hour for operation. The Newview 100 and RST-Plus are' not easily accessible and
cost approximately $300 per hour for operation. The ESEM is also easily accessible, but
cost $40 per hour. In comparing all the instruments, the Dektak can give a good
approximation of the thickness at the edge of the diamond. However, the RST-Plus gives
the best and most complete results because of its ability to show how uniform the
diamond thickness is at all locations with an uncertainty of about 0.05 microns.

5.6 TEMPERATURE ACQUISITION

The acquisition of temperatures involves capturing thermal images of the
diamond window in steady or quasi-steady state, and then extracting the desired
temperatures, in this case along a 3mm line. However, before the images can be
captured, it is necessary to properly align the sample in the desired ﬁeld of view because
the 3mm diamond window on an opaque surface can be misread easily when viewed
through the telescopic zoom lens. The process of aligning and focusing the 3mm
diamond window is a difﬁcult and time consuming task. Once the diamond window can

be viewed properly through the telescopic zoom lenses, a calibration must be made to

relate physical distance on the sample to radiometer distance as seen on the RGB
monitor. The procedure necessary to accomplish these tasks is explained in the following

paragraphs.

The setup of the sample to capture thermal images is carefully performed since
handling can easily damage the diamond window. First, the sample is placed between the
jaws of an adjustable clamp. The jaws are then tightened so that the sample is immobile
and suspended. The upper left or right comer is the only area where the sample is in
contact with the clamp. This positioning is convenient because the electrical tape at the
bottom of the sample must hang for easy access to the electric lead wires. The clamp and

sample (shown in Figure 5.30) are then placed on a ﬂat vertical motion jack.

 

Figure 5.30 Adjustable clamp with sample

88

This jack was used because positioning the sample in the deSired ﬁeld of view requires
many movements of the sample in the horizontal and vertical directions, Therefore the
ﬂat surface allows for easy horizontal motion of the clamp and the adjustable nature of
the jack allows for easy vertical movements. These precautions are necessary because
extensive movement (and handling) of the sample might lead to destruction. Care must
be taken to assure only a small portion of the sample is in contact with the clamp,
otherwise heat conduction might occur through the clamp and ultimately affect the

experimental model.

Once the sample is suspended in front of the scanner, the next step is to relate
precise locations and distance as imaged by the scanner to those of the actual sample.
The relation was deﬁned by using a machined thin aluminum plate with a 3mm open
area. The aluminum plate was then placed in the same ﬁeld of view as the sample where
the temperature of the open area was elevated for clarity. Two procedures were used to
identify the 3mm region. One technique was to use the electronic ruler and coordinates
(on the image processor) to identify the number of pixels equivalent to 3mm. This really
involves identifying the silicon/diamond interface. This procedure involves identifying
the silicon/diamond interface. Another technique was to record the image corresponding
to 3mm on videocassette using the VCR (Panasonic AG—2400). Recording the image is

important if other methods (such as image processing software) are to be used to extract

89

temperatures from the thermal images. Once the calibration scale has been determined,
the control unit is set to the desired settings (emissivity, image average, and temperature
span, etc.). The electrical wires are then connected to the sample, and a constant voltage
is applied. The temperature of the sample increases immediately, and the image of the
sample in quasi-steady state is captured. The shape proﬁle or temperature gradient across

the 3mm diamond window is the same in quasi-steady state as it is for steady state.

Two methods were used to capture the diamond window in quasi-steady state.
One method involves using the freeze frame function on the image processor. This
involves simply freezing the image and subsequently saving the still thermogram on disk.
The other method is to record the entire heating process on video, and later capturing the

desired image on playback.

After the thermal images have been acquired using the appropriate control setting
on the radiometer, the temperatures can be extracted using the File_Access program run
by the gwbasic interpreter with the ThermoGram card or the software Image Pro Plus
(IPPlus). The File_Access program, which is part of the thermal image processing
software, extracts data from a particular line of the thermal image and converts the
individual pixel intensity to temperatures using calibration tables stored in the ROM of
the radiometer’s microprocessor. The program prompts the user for the ﬁlename of the
thermal image and the desired line number within that image. The temperatures

corresponding to the pixel location across the entire image are then written to a data ﬁle.

90

Another method to extract temperatures is to use the soﬁware Image Pro Plus with a VCR
and the recorded heating process. Since IPPlus is not a thermal image processing
software, it must ﬁrst be calibrated in order to convert pixel intensity to temperature. A
calibration scale corresponding to the temperature span of the radiometer was used.
Therefore, once the 3mm region is known (through the use of the electronic ruler or
recorded calibration scale), the corresponding pixel intensity (and then temperature) can
be extracted by simply selecting the 3mm line corresponding to the diameter of the
diamond window. Two methods were used because equipment failure by the
ThermoGram card forced the use of IPPlus. Therefore, experiment performed in the
latter phase of the project were processed using IPPlus. Samples #1,2,3 and 17
experiments were processed using IPPlus, while samples #19,7,8,6, and 4 were processed
using the ThermoGram card. With the desired temperatures obtained, it is now necessary

to estimate the thermal conductivity of the sample.

5.7 DATA PROCESSING AND ESTIMATION OF THERMAL
CONDUCTIVITY

Parameter estimation is a discipline that provides tools for the efﬁcient use of data
in the estimation of constants appearing in mathematical models and for aiding in

modeling of phenomena (Beck, 1977).

With the appropriate temperature distribution and the governing equation, the
program NLIN (design by Dr. Beck) is used to analyze the data and estimate the thermal

conductivity. The FORTRAN program NLIN uses the method of least squares

9l

N

S = 20: -T.-)2 (5.3)

where N is the number of temperature measurements. Y, is the measured temperature, and

T, is the calculated temperature represented by

 

T. = T. + HKfI-(Z—‘Yl (5.4)
2 2
where H: V b , K = l
4RLW(6d +63) k

and V=applied voltage (volts), b=radius (meters), R=resistance of gold coated sample
(ohms), L=length of circuit (meters), W=width of circuit (meters), 8d=thickness of
diamond (meters), ﬁgr-thickness, of gold (meters), k=thermal conductivity (Watts per

meter Kelvin), Tb=temperature at the boundary.
Using the method of least squares, NLIN simultaneously estimates Tb and k to calculate

T, such that the error in S is minimized. In order to minimize the error in S, NLIN uses

the system of equations

5' N
E-QZW-m-O (5'5)

92

68 N ’3‘ 2 _
E’"ZZ(’7‘ ,)H(l—(;))—0 (5.6)

While these equations form the basis of the linear estimation problem, NLIN is actually a
sequential nonlinear estimation program with more complex calculations than those
shown above. Not only does NLIN ﬁt the experimental data to the mathematical model,
but it also calculates the residuals which are the difference between the measured and

calculated temperatures (Y, - Ti). The estimation of K and the calculation of the residuals

2 2
are carried out with the input H = V b , initial guess of Tb and K = ~1- , and
4RLW(6,, + 5:) k

 

the temperature distribution (Herr, 1993). Typical input and output ﬁles are shown in the

appendix.

93

CHAPTER 6

RESULTS AND DISCUSSION

The techniques and procedures discussed in the previous chapters were used to
acquire accurate temperatures and estimate the thermal conductivity of each sample.
There were a total of 9 samples. All samples were fabricated in a similar manner,
therefore the only signiﬁcant difference from sample to sample was the level of doping as
measured by the electrical resistivity. The purpose of the research was to investigate
what effects the measured resistivity (doping level) has on the thermal conductivity of
boron doped diamond ﬁlms. In this chapter, the results of the experimental
measurements are shown, along with a discussion of the factors which contributed to each
sample measured thermal conductivity. The measured thermal conductivity and electrical
resistivity are presented with experimental uncertainty to indicate the conﬁdence (or

certainty) to which the results are known.

6.] THERMAL CONDUCTIVITY AS A FUNCTION OF ELECTRICAL
RESISTIVITY

The estimation of the thermal conductivity was performed using the program

NLIN. Representative results of an experimental test for sample #17 showing

94

temperature distribution, residuals, and sequential estimation is given below. A complete

set of results for all samples is given in Appendix A.

T0mp0r0tur0 (C)

 

19

rib

Figure 6.1 Measured versus calculated temperatures (Sample #17)

Figure 6.1 shows a plot of the measured temperature (diamond symbol) across the 3mm
diamond window, along with the calculated temperatures as predicted by the simpliﬁed
governing equation (solid line). From the ﬁgure, its easy to see that the temperatures
match fairly well except for measurement noise inherent in the infrared thermograph
system. A better indication of how accurate the measured temperature is compared to the

calculated temperature is shown in Figure 6.2. This ﬁgure is the residuals (Yr-Ti), and it

95

represents the calculated temperature minus the measured temperatures at each location

along the 3mm diamond window.

 

0.5 ..

.4 _ .11 _______

0 — 1 20' '30" 40 50" :O‘ 87;.0- ‘1 “IA—B0 . 1%

Residuals (C)

05 .

 

 

 

Location across 3mm window

Figure 6.2 Residuals (Sample #17)

Figure 6.2 shows a noise level of about 0.5 °C. This measurement is fairly accurate
considering that a noise level of 0.25 °C was measured at room temperature and is the
best accuracy to which the infrared camera can measure the diamond surface. Also, a
noise level of 0.5 °C is small when compared to the maximum temperature gradient of
about 10 degrees from r=b to r=0 across the diamond window. The residuals are
calculated for each measurement from r=-b to r=+b, therefore data point #1 corresponds

to r=-b (-1.5mm) and data point #93 corresponds to r=+b (+1.5mm).

96

400 __ __,___m--- _.__, -HM.-.-._.,-.-L-,,W _ . -.u__ _,,

350

300 .4.

250 .-
2081165453

 

 

J

150 ..

Thoma) Conductivity (WI m K)

 

 

0 10 20 30 40 50 00 70 BO 90 100
Location across 3mm window

Figure 6.3 Sequential estimation of thermal conductivity (Sample #17)
The measured temperatures are then used to estimate the thermal conductivity. Figure
6.3 shows the sequential estimate as each data point is added. It’s easy to see that with
only a small amount of temperature measurements the estimate ﬂuctuates. However, as
more temperature measurements are considered an asymptotic value is approached. In
this case, thermal conductivity is estimated to be 208.1 W/m K. The estimate was
performed from r=-b (-1.5mm) to r=+b (+1.5mm), and the data points are numbered

similar to that for the residual data.

The overall results show that thermal conductivity increases as doping level

decreases (electrical resistance increases). Figure 6.4 shows that there exists an

97

“exponential” type relationship between thermal conductivity and average electrical

resistivity.

'"958 .-. -. ,. .., , .. '975

750 - 740

650 . 553

561

WConduetivityMlmk)
{a §ﬁ§

200.. 205

 

 

0 5 10 15 20 25 30 35 40 45 50 55 60 65 7O 75 80 85
midwity (011111 0111)

Figure 6.4 Thermal conductivity as a function of average resistivity

Based on the distribution of boron powder which substitute into the diamond lattice as
diamond crystals nucleate from the surface of the silicon, and the notion that samples
close to the boron source is exposed to a higher concentration of boron atoms, the results
shown in Figure 6.4 is consistent with doping levels and it’s effect on diamond. The
above graphs indicate that samples with high levels of boron doping (low resistivity)
posses relatively low thermal conductivity, while low doping levels (high resistivity)

samples posses relatively high thermal conductivity. Also present in the “exponential”

98

nature of the curve, which has a steep slope at low resistivity values which subsequently
decreases as resistivity increases, is the approach of a limit. This would indicate that a
particular value of thermal conductivity is being approached as the resistivity increases to
inﬁnity. Such trends imply that the effects are entirely due to the addition of boron
during synthesis, and had synthesis been carried out without boron (inﬁnite resistance) an
average value consistent with inﬁnite resistance would be measured for all samples
considered. Likewise, had an inﬁnitely large amount of boron been used, it is presumed
that the thermal conductivity would approach that of boron which is about 27 W/m K
(Kittel, 1996). One method commonly used to characterize the doping level is to use a
Secondary Ion Mass Spectroscopy (SIMS). This was done by Malta, Windheim, and Fox
(1986) and it was discovered that polycrystalline boron doped diamond at 300K with a
measured resistivity of approximately 600 (2 cm posses atomic boron concentration of
2x1018 cm'3. If it is assumed that there exist a linear relationship between atomic
concentration and resistivity, it can be concluded that samples used in the present work
with average resistivity which range from 1.63 to 80 (2 cm posses boron concentrations at
least 7.5 times greater than the Malta et al. sample. In order to ﬁrlly understand the
relationship between atomic boron concentration and the measured resistivity reference is
made to a 1996 paper by Werner et al. In their work a four point probe was used to
measure resistivity and a Secondary Ion Mass Spectroscopy (SIMS) was used to
determine the amount of boron atoms present. In Figure 6.5 below (Irvin Curve), the
relationship between resistivity, Hall measurements, and effective doping concentration is

given for single and polycrystalline diamond.

99

 

 

 

 

k9". ..1 ' 10a 10‘
e818t1v1ty (Dem)
Figure 6.5 Irvin curve for diamond (Werner et al. 1996)

Circles are for single crystal and triangles are for polycrystalline diamond, with open
symbols representing effective doping concentration, N A - ND , and ﬁlled symbols
representing free hole concentration. From the ﬁgure, it was concluded that the current
mechanism depends on the doping level and a simple room temperature resistivity
measurement is required to estimate the doping level. Based on the above graph, it is

possible to estimate the amount of boron for local resistivities used in this work. Table

6.1 gives an estimate of the doping level for all samples.

100

 

 

 

 

 

 

 

 

 

 

Local Resistivity Estimated Atomic Concentration

(:2 cm) x 10 ‘9 cm'3
1.63 8.0
2.2 6.0
5.51 3.5
6.73 3.0
9.82 2.9
14.96 2.3
21.85 2.0
28.38 1.6
68.43 1.1

 

 

 

 

Table 6.1 Estimated doping level for all samples

One difﬁculty commonly encountered in using boron powder as the doping source
in diamond synthesis is the variability of resistivity over relatively small areas of each
ﬁlm. For this reason, the above graph of thermal conductivity is plotted versus average
resistivity. Another way of representing the results is to plot thermal conductivity versus
local resistivity. The local value of resistivity was measured in the center of the sample
prior to etching (i.e. in the etched region where temperatures were measured during
experimental runs). This was done to investigate if there exist signiﬁcant differences
between the local region of interest and the overall expected value of ﬁlm resistivities.

Figure 6.6 shows the thermal conductivity vs. local resistivity (measured at the center).

l0]

 

 

 

 

° 97§
960 .
S?
5'
50,.
o in
o 5 1o 15 20 25 30 35 40 45 so 55 so 65 7o

mummy (ohm cm)

Figure 6.6 Thermal conductivity as a function of middle resistivity

Comparison of the thermal conductivity graphs (Figure 6.7 combine plot of average and
local resistivity) indicates little variation, therefore it can be concluded that conductivity
of each sample is unique to the range of resistivity measured using the four probe method

and the overall shape of the thermal conductivity-resistivity data does not change.

W Conductivity (Win 1:)

300+ 5..

2001

100.

 

10

102

a;;. - __...._..- ._ ,4

4O 50 60 7O 80 90
Roddvlty (ohm cm)

Figure 6.7 Combined plot of measured thermal conductivity

Another important point in the measurement of resistivity of each ﬁlm is that while there

were variations within each sample, there was no overlapping among the measured values

for different samples. The non overlapping indicates that each sample possesses a unique

range of resistivities. Along with knowing the range of resistivities for all samples, it is

necessary to understand the variability within a given sample. The standard deviation of

the measured resistivities within each sample was calculated and shown in Table 6.2

103

 

 

Sample # ieaauremente Average reelettvlty (ohm cm) §'tandard Devlatlon Coet'Tlclent of Variation

17 8 1.621 0.335 0.217
" 19" ‘ '4' 3.96 ' ' 2.366” " 0.599
T 8' ' " 3? 6.765' 1.22' 0.180

7 5' 6.065' 1.342" ' 0.166

6 3' 11.66' " 2.606 0.240
'3 3' 3 3' 13.396“ 3666' ' " 0199
F" “'4'" ' 3 ' " 26.53 " 5.346 ' 0.205
"" 3 ' ' ' 3' 31.676" ' ' 36.546 3 0.207'
— 3T 3 3 ' 3' ' '80.'1'4'2""' '3 10.18 ' 3 0.127

 

 

 

Table 6.2 Average resistivity and standard deviation
The estimated values of thermal conductivity shown in Figure 6.4 and Figure 6.6
are average values for multiple experimental runs. The experiments were performed
under similar conditions except for slight variation in power setting which has shown no
effect on the estimated conductivity. While the average value gives some indication of
the expected value, the variability of the estimates must also be understood. For this

reason, the standard deviation of all estimates used to calculate an average value is given

in Table 6.3.

' WhTIOTWT Tl: 3" ‘ 90704 T "0.09” _*_OEOT”_W_O'.OB'*TIIUS' TO:

 

Table 6.3 Estimates of thermal conductivity with average and standard deviation

The data from Table 6.3 indicates that as the measured value increases so does the
standard deviation. This trend implies that samples with low thermal conductivity can be

measured with a great deal of precision. However, as the estimated thermal conductivity

104

increases, the standard deviation decreased. Overall, the coefﬁcient of variation (standard
deviation divided by the mean) for all samples measured were small (0.01 to 0.09) and

seem to be independent of the average resistivity.

In order to gain insight into the thermal behavior of boron doped diamond ﬁlms
and the results presented, it is necessary to examine the impurities introduced into the
now electrically conducting diamond. Impurities in a semiconductor can scatter electrons
and phonons in non-metallic crystals such as diamond. Impurities, which are brought
about by the boron, can also introduce free electrons and holes, and these can reduce the
phonon conductivity by increasing the scattering. If the concentration of holes or
electrons is great enough, they can contribute an appreciable component to the total
thermal conductivity (Berman, 1976). In order to simulate the measured thermal
conductivity as a ﬁmction of doping level (as measured by the resistivity), a comparison
between the thermal conductivity of perfect single crystal diamond and boron doped
polycrystalline diamond is investigated. The theoretical relation which governs thermal

conductivity in single crystals is given by (Berman, 1976):

l9

k k T x4e“r
k=——L —3 3T3 ———dx 6.1
2791/92) gm)“, -1)2 ( )

where x= h m/kBT

105

and

h - Planck constant, k3 - Boltzman constant, O - Debye temperature (2230 K),
T - Experimental temperature, 0) - Phonon frequency, T(X) - Total scattering rate,
and v - average phonon velocity (speed of sound). Equation 6.1 is the Debye model for
thermal conductivity. The Debye approximation works well because even pure crystals
are usually just not perfect enough for N processes to be very important (Berman, 1976).

Therefore, only resistive processes are considered in this analysis.

In the Graebner et al. work (1994), thermal conductivity of diamond was
calculated using a combination of scattering processes. In Figure 6.8 below, the
uppercurve is obtained using only boundary scattering and umklapp scattering. The
curve labeled a+b is the same but with thickness, (1, equal to 1.5um. The remaining
curves are calculated with additional scattering by point defect (p), dislocation (5),
extended defects (e), and microcracks (m). The graphs shows that thermal resistance is
dominated by point defects, while extended defects, dislocations, and microcracks play a

lesser role.

lO6

((Wcm ' K ')

 

 

T (K)

Figure 6.8 Graebner et al. results showing effect of each type of defect on thermal
conductivity (Graebner, 1994)

Graebner et al. (1994) and Morelli et al. (1988) examined diamond ﬁlm thermal
conductivity over a range of temperatures using the theoretical relationship in order to
quantify what effect scattering processes have on the measured thermal bonductivify.
While this approach cannot be used in the present study to analyze the measured values
1 since all experiments were performed at about 10 °C above room temperature and the
microstructural analysis necessary to measure grain size, etc. was not performed, the
analysis can be carried out to verify whether the functional relationship given above (and
used by Graebner and Morelli) is valid. Assuming that the total scattering rate is
dominated by resistive processes, the governing equation can be simpliﬁed. With the
governing theoretical equation a function of all scattering processes, it is now possible to

simplify the model even further by assuming that point defects, brought about by boron

addition, is responsible for the variation in measured thermal conductivity. Consider

107

scattering processes due to crystal boundaries, point defects, dislocations, and extended
defects. A point defect is a defect which extends over a volume with linear dimensions
much smaller than the phonon wavelength. At a given temperature the dominant phonon
wavelengths for thermal conduction are on the order of a few interatomic distances (the
exact value depends on the scattering law). The defect conﬁned to a few atomic volumes
will be small enough to appear as a point defect to the important phonons. A defect
which ﬁts this criteria may be another atom (such as Boron) on a lattice site substituting
for the correct atom (carbon), a vacancy at a lattice site, an interstitial atom, or a
combination of these. The scattering is then caused by the difference in mass and
difference in bonding between the atoms (Berman, 1976). If a defect is larger than a few
interatomic spacing, the scattering does not follow the same mathematical equation as a
point defect and must be considered separately. Defects of this kind are referred to as
larger or extended defects. A dislocation consists of a narrow region along its axis within
which there is a drastically altered structure which may be represented by a change in

density (Berman, 1976).

Taking into consideration that the ﬁlms were created under similar conditions
except for the variation in boron levels, its reasonable to assume that all resistive
processes are constant for all ﬁlms except for processes due to point defects. Total

scattering is then given by

r(x)" = rB(x)" + 1,, (x)" + 1,. (x)" + 1,,(x)" (6.2)

108

where 13(x) - scattering due to crystal boundaries, tD(x)- scattering due to dislocations.

TE(X) - scattering due to extended defects, and rp(x) - scattering due to point defects.

TI! TI) TI; Tr

 

rewritten as r(x) = ( (6.3)

TI) 71; + TB T]: + TB TI) )71’ + T3 TI) TI;

C 1'
this equation for total scattering rate can be simpliﬁed to r(x) = 31%.— (6.4)
2 3 P

where CI = 1,3101”, C2 = 13101, , and C3: 1015 + 1315 + 731,,
With these effects (scattering due to dislocation, extended defects, and grain boundary)

considered constant for each ﬁlm, the governing equation can be simpliﬁed even further

 

to
C r
k = —‘—”—— (6.5)
C2 +C3r,,
0
where C4 = 5:)?"3 I———dx
27231!

and 1,, is related to the concentration of point defects. This form was developed using the
following assumptions:

1) All scattering mechanisms are independent of frequency (and therefore wavelength).

2) The ﬁlms are imperfect and all modes are strongly scattered by resistive processes.
Therefore in the Callaway expression for thermal conductivity K. >> K2.

3) N processes are neglected.

109

If the above analysis is correct then this is the functional form for thermal conductivity as
a function of point defects due to boron substitution into diamond’s lattice structure. A
curve ﬁt of experimental thermal conductivity as a function of local resistivity is shown

in Figure 6.9.

1200 .... .. . - ..

___,__,.._...vo,_—_———«
‘ "V"...—

1000 . if-”

.—
,1
o"

'l'hem'lal Calduwvlty (Win It)
a

 

 

0 10 20 30 40 so 60 70
Reddvlty (ohm cm)

Figure 6.9 Curve fit of thermal conductivity as a function of middle resistivity
The software Tablecurve 2D by Jandel Scientﬁc was used to analyze the data and
produce functional equations. For the data, the equation

b
k" =a+-, (6.6)

x

where a and b are constants, k is thermal conductivity, and x is concentration of boron
(as measured by the resistivity), match the functional form given above with a correlation

coefﬁcient of 0.966. This is evidence that the functional relation given by the theoretical

”0

equation (and used by Graebner and Morelli in their analysis of experimental thermal
conductivity) governs the experimental work in this report. Since results are presented
for local and average resistivities, it is useﬁJl to curve ﬁt average data as well. The curve

ﬁt of thermal conductivity versus average resistivity is shown in Figure 6.10.

I
,o
'1
I

Thermal Condodivity (WI!!! K)
5 f5
0

.§

 

 

0 1O 20 30 4O 50 60 70
a eelellvlty (ohm cm)

Figure 6.10 Curve ﬁt of thermal conductivity as a function of average resistivity

For thermal conductivity as a function of average resistivity the correlation coefﬁcient
was 0.937. Once again the curve ﬁt matches fairly well considering the microstructural

analysis necessary for all scattering processes to be understood and quantiﬁed was not

performed.

111

6.2 UNCERTAINTY ANALYSIS

Error may be deﬁned as the difference between the measured result and the “true”
value. Because the true value is never known, its difﬁcult to know the error. However, it
is possible to estimate an error. The process of assigning a value to an estimate of error is
commonly referred to as an uncertainty (Beckwith, 1990). In determining the thermal
conductivity of thin diamond ﬁlms, many values were measured. Each of the measured
values contain errors which must be accounted for. Since the measured values are
subsequently used to determine the magnitude of the thermal conductivity, the

uncertainty in the estimated thermal conductivity must be calculated.

The uncertainty in the thermal conductivity is determined using the estimated error for
the measured parameters and the simpliﬁed governing equation written in terms of

thermal conductivity as

V2b2 7'2

= 4RLWO(5d + 519(1- 32—) (6‘7)

 

K

The uncertainty in K acts a global uncertainty which is driven by the precision with
which all experimental unknowns can be measured (Herr, 1993). The uncertainty in the

thermal conductivity is determined ﬁ'om the expression

ll2

_ 2‘1: 2 a: 2 ﬂ 2 11 2 ﬁ 2

[AK—[(WAV) +(o’RAR) +(5LAL) +(0‘WAW) n+(a9A0) (68)
.295. 2 a: 2 a 2 2K. '
+(ajAb) “36 A68) HOE, A6d) +(07Ar) ]2

8

where

5K __ Vzb2 _ r 2
517' 2RLW6(6d+6g)[l (b) 1 (6'9)

 

0K Vzb2 r
__=- 2 [1—(—)2] (6.10)
o‘R 4R LW6(6d + 68) b , .

 

0K Vzb2 r 2
Z-—4RL2W9(5d+58)[1-(3) ] (6'11)

 

07(__ Vzb2 _ i 2
35' 4RLW192(64+68)[1 (b) ] (6'12)

 

0K __ Vzb2 _ I. 2 '
2‘17- 4RLW26(6d+6g)[1 (13)] (6'13)

 

 

37g V26
5b

= 2RLW9(6d + 522) (6'14)

113

aK Vzb2 r 2
—=- .[1—(—>-1 (6.15)
66‘, 4RLW6(5,,+6X) b

 

o‘K V262
_=_ 2 [1—(5)2] (6.16)
0‘6 4RLW19(6,, + 5,) b

8

 

 

5K V2
5- = r (6.17)

' 2RLW6(6d +52)

and AL=1><10'5 m, Aw=1x10'5 m, AR=lx10'3 Q,AV=1x10'3 V,A0=0.1°C,A5d=lx 10'7
m, A8g= 1x 10'9 m, Ab=lx 10'6 m, Ar=lx 10'6 m. Since various power levels (P=V2/R),
lengths, widths, etc. were used to estimate the thermal conductivity, the uncertainty in
thermal conductivity varies for each sample and for each experiment (due to various
power settings). Because of the varying uncertainty, the analysis was performed for all
conditions, however only worst case or highest value in the estimate of the errors are

presented in this section.

The uncertainty in the measured parameters were estimated based on the precision
of the instrument used. The uncertainty in the voltage was chosen as 0.001 volts because
this is the precision to which the digital multimeter (DMM) can measure voltage. The
same DMM was used to record sample resistance (not the diamond resistance as
measured by the four probe method), and did so with a similar uncertainty of 0.001 (2.

The measurements of length, width, and radius were performed using a vernier caliper

114

which was capable of measuring to one hundredth of a millimeter. Therefore, the vernier
caliper presented measurements with uncertainty of 0.00001 m. The uncertainty in
temperature was chosen as 0.1 °C due to the sensitivity with which the infrared system
measures temperature. The uncertainty in the thickness of the gold and diamond were
conservative estimates based on the instruments used to sputter gold and measure
thicknesses in each case. The uncertainty in the thickness of the gold was based on the
manufacturer’s calibration of sputtering rate. The uncertainty in the thickness of the
diamond was chosen to be 0.1 microns based on the Dektak measurement precision used
and the assumptions made. This is a very conservative estimate with a relative error of 5
percent. The estimate of each parameter uncertainty and the equation for uncertainty in
thermal conductivity was then used to calculate worst case global uncertainty of thermal
conductivity for all samples measured. Table 6.4 shows the uncertainty of each sample

and the percent error associated with the measured value.

 

Table 6.4 Uncertainty in measured thermal conductivity

The % uncertainties are within the range reported by other researchers. In Herr’s (1993)

analysis of 3 doped diamond ﬁlm, the thermal conductivity was found to be 249i13 W/m

115

k. This analysis give a percent uncertainty of 5.22 which is close to the 6.0% average in
the present work when all samples are considered. Comparing the measured standard
deviation (Table 6.2) to the calculated uncertainty (Figure 6.11), it is easy to see that

samples #1, #2, #3 and #17 have signiﬁcantly different values for standard deviation and

 

 

calculated uncertainty.
70 r --
2.2 i
so i
55 _
50 «L
45 ..
g .. _.
5 35 .
3 2. l
25 l
20 l
15 2
10 ..
5 2.
0
O 5 10 15 20 25 30 35 4O 45 50 55 60 65

 

Standard Devlalon

Figure 6.11 Comparison between calculated uncertainty and measured standard
deviation
One possible explanation for the small standard deviation could be that these samples
were estimated based on temperature data processed using IPPlus. Similar results
showing small relative deviation for samples #1, #2, and #3 are shown in Figure 6.12,

which is a plot of standard deviation versus thermal conductivity.

116

 

SW Deviation
8
8

20.00 ..

10.00 I

 

0.00

 

200 300 400 500 600 700 800 900 1000
Thermal We!” (Winn K)

Figure 6.12 Measured uncertainty relative to mean thermal conductivity estimates

The complete uncertainty analysis has the advantage of revealing the contribution
of each parameter to the overall uncertainty. Table 6.5 shows the magnitude to which
each parameter contributes to the overall uncertainty in the estimate of thermal
conductivity. Based on this information it can easily be seen that the thickness of the
diamond and the sensitivity of the temperature difference are responsible for the error in
the estimate of thermal conductivity. Therefore, in order to reduce the experimental
uncertainty, accurate measurement of temperatures and diamond thickness’ are crucial.
While experimental limitation (such as sample size, etc.) require the use of an infrared

system to record temperatures, there is certainly room for improvement in the

1 l7
measurement of the diamond’s thickness. One way to improve the uncertainty is to
examine the sensitivity of thermal conductivity to thickness of diamond. Figure 6.13

shows the estimated thermal conductivity for a range of thicknesses for sample #2.

1100 4

1000

Estimated mammal Conductivity (WIm K)

 

 

500

 

1.5 1.6 1.7 1,8 1,9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3,5
“icicle-(mime)

Figure 6.13 Sensitivity of thermal conductivity to thickness of diamond
It can easily be seen that thermal conductivity becomes less sensitive to thickness as the
magnitude for thickness increase. For this reason, Herr (1993) and others measured

thermal conductivity with thicker samples.

118

 

 

 

 

 

 

 

 

 

Parameter, x Contribution % Contribution
dK 2
( dx dx)

Voltage, V 2.39 0.067
Resistance, R 0.38 0.011
Radius, b 38.09 1.054

Length, L 1.09 0.03
Width, W 0.46 0.013
Temperature Difference, 0 0.46 44.87
Gold Thickness, 5,, 1620.22 0.135
Diamond Thickness, 8,, 1943.58 53.823

 

 

 

 

 

Table 6.5 Parameter contribution to uncertainty in thermal conductivity

In order to quantify the results presented (thermal conductivity as a function of
resistivity) earlier, the uncertainty in the measured resistivity must also be calculated.
Once again conservative estimates of the parameters were utilized so worst case

uncertainty can be calculated. The uncertainty in resistivity is given by

Ap= [(gj-AW a; A1)2 «£402 +121; 225)]: (6.17)

where

__=c_ win

119

if = — —V,— (6.19)
ﬂ a-

£2 = (SE (6.20)
or: 1

53.8 = CK (6.21)
55 1

and AV=1 x 10'3 v, Al=0.l A, AC: 0.1, and A6=1 x 10'7 m.
The above equations were used to calculate the uncertainty in resistivity. The magnitudes

of the uncertainty are shown in Table 6.6 and 6.7, along with percent error.

Ol’l'Of

 

Table 6.6 Middle resistivity, uncertainty, and percent uncertainty

 

Table 6.7 Average resistivity, uncertainty, and percent uncertainty

120

average

 

 

 

 

 

 

 

 

Parameter, x Contribution % Contribution
Voltage, V 0.007 68.62
Constant, C 0.003 29.41

Current, 1 0.0004828 4.73

Thickness, 5 0.000001207 0.01

 

Table 6.8 Parameter contribution to uncertainty in resistivity

Table 6.8 shows the signiﬁcance of each parameter on the overall uncertainty.

Another way of presenting the uncertainty is with the use of error bars. Error bars

has the added advantage of showing the range of error relative to the scale of all estimates

graphically. Figures 6.14 and 6.15 shows a graph of thermal conductivity versus

resistivity with error bars.

 

121

ThemalconduotivhthnK)
“ﬁg“???
"37'

 

 

0510162025303540465055606570756065
laddvltyphmcm)

Figure 6.14 Thermal conductivity as a function of average resistivity with error bars

 

A L 1
v v

TherdeonducttvhyMlmK)

oséégggggrgsgg

L J; A
Y

 

 

10 20 30 40 50 60 70 60 90
mm (ohm cm)

0

Figure 6.15 Thermal conductivity as a function of middle resistivity with error bars

122

It can easily be seen that the magnitude of the error increased as the magnitude of the
measured quantity increased. However, the percentage or relative error did not change

signiﬁcantly.

123

CHAPTER 7

SUMMARY AND CONCLUSIONS

The goal of this research was to determine the effects of boron doping on the
thermal conductivity of diamond ﬁlms. In pursuit of such a goal, an experimental
technique was used which allowed for electrical resistance heating for a range of doping
levels. The preceding chapters present a comprehensive method for the fabrication and
thermal analysis of boron doped CVD diamond ﬁlms. The basic techniques and
procedures necessary for the growth of diamond using the hot ﬁlament CVD method is
described, along with the required sample design for the radial heat ﬂow model, and then
the thermal activation and capture of the temperature distribution for the estimation of

thermal conductivity using the methods of parameter estimation.

In accomplishing the goals of the project, thin diamond ﬁlms were deposited on
silicon wafer using the hot ﬁlament chemical vapor deposition method. Doping was
made possible by placing boron powder in a honeycomb shape holder on the silicon
wafer. A mask was used in order to have many diamond ﬁlms with various doping levels

on a single 4 inch silicon wafer. The HFCVD system deposited diamond for 10 hours at

124

a rate of approximately 0.2 microns per hour. The silicon wafer was then cut along the
mask regions in order to separate each ﬁlm from the bulk wafer. The doping level, as
measured by the ﬁlm’s resistivity, was then measured using the four probe method.
Thickness measurements were then made using the Sloan Dektak IIA proﬁlometer before
a 3mm diameter of free standing diamond was created by chemically etching the silicon.
The etched sample is prepared by forming an electric circuit with the diamond. In order
for the infrared thennograph system to measure accurate temperatures, a thin layer of
gold is added to one surface of the diamond. The experiment is carried out by passing an
electric current through the diamond. The internal heat generated by the current causes
temperature gradient within the 3mm free standing diamond. The radial heat ﬂow model
and the measured temperatures are then used to estimate the thermal conductivity. Each
sample was fabricated in a similar manner, and the estimated thermal conductivities were

averaged for several experimental runs.

The results of the investigation revealed that boron doping decrease the thermal
conductivity (when compared to undoped diamond). This result is predicted by theory
since the addition of boron to the diamond structure brings in defects in an otherwise pure
material. The defects in the diamond is responsible for the reduced thermal conductivity
measurements, and the relative amount of defects present is responsible for the magnitude
of the thermal conductivity. Because of the complex nature of the theoretical equation
governing thermal conductivity and the molecular structure responsible for heat

conduction in diamond crystals, it was not possible to compare calculated theoretical

125

thermal conductivity to measured values. However, the functional form of theoretical
thermal conductivity was compared to the functional form of the experimentally
determined thermal conductivities. This analysis revealed that resistive processes is
responsible for the measured conductivities. However, its important to note that the
dominant process is point defects. Since all samples were fabricated under similar
conditions, its reasonable to assume that extended defects, boundary scattering, and
dislocations would be constant for all samples. Point defects is not constant however,
since the boron source contributed varying amounts of boron atoms to substitute into the

diamond lattice.

While the thermal conductivity is given as a function of average and local
resistivity, it should be noted that local resistivity gives a better indication of the amount
of boron present in the region of interest. It is within the etched region where
experiments are performed and temperature distribution are extracted for use in
estimating the thermal conductivity. For this reason, the local resistivity versus thermal

conductivity gives a better ﬁt to the theoretical equation as a function of point defects.

Lastly, since little information was found in the literature related to thermal
analysis of boron doped diamond ﬁlm, it was not possible tocompare the results to other
experimental studies. For this reason, this work is an initial study and will aid future
studies on the thermal properties of boron doped diamond ﬁlms and all related

applications.

126

CHAPTER 8

RECOMMENDATIONS FOR FUTURE WORK

The present work describes an experimental technique and results for thermal
conductivity of boron doped diamond ﬁlms. As in most experimental work there is
always room for further study in order to minimize the uncertainty and improve the
overall wealth of knowledge in the ﬁeld of study. In this section, some of the necessary
procedures which can be used by future research in the study of semiconducting diamond

ﬁlms will be addressed. The recommendations will include the following:

9 Development of a technique which can control the level of doping. This technique
might include depositing diamond samples one at a time using a solid or gas form of
boron. The mass of boron could then be adjusted in order to acquire the desired
resistivity. While a solid source can still lead to varying resistivities over small
regions, a gaseous source might not. However it should be noted that little is known

about boron distribution when boron powder is used as the doping source.

0 Development of an accurate and non-destructive method to measure ﬁlm thickness

and thickness proﬁle.

Check the accuracy of the technique using a known material or a different technique

with the same samples.

Expand the range of resistivities and include more samples.

Examine samples on the microscopic level in order to investigate defects. This could
include using Raman spectroscopy, Scanning electron microscope, or other

instruments which can quantify the presence and abundance of defects.

Use a Secondary Ion Mass Spectroscopy (SIMS) to quantify the amount of boron
present. This will permit determining the thermal conductivity as a function of the

amount of boron atoms present which can be related to measured resistivity.

128

ABBENDIXA

EXPERIMENTAL DATA FOR EACH SAMPLE

SAMBLEEIZ

SAMPLE £17
Meaeured ve Calculated

 

 

Temperature (C)

 

 

—e— $00031
—— Series2

 

 

FIGURE A.l

-O.6

rib

129

SAMPLE #17
RESIDUALS

 

—e— $011031

Temperature (C)

41.4

 

 

Number of data pointe

FIGURE A.2

SAMPLE I17
Estimation of Thermal Conductivity

 

200162899

— Setieai

 

Thermal Conductivity (win 11)

0 10 20 30 40 50 60 10 80 90 100
Number of data points

FIGURE A.3

130

SAW

SAMPLE #8
Measured ve Calculated

38 ._

 

'—'3

Temperature (C)

 

 

+ Sen'esi
—. Set-16:2

 

FIGURE A.4

 

131

SAMPLE I8
RESIDUALS

0.6 ., .. -W--. - -_ -1 -W

 

0.4 .

0.2 2

 

120
—e— $011081

-0.2 .

Temperature (C)

 

'0.‘ J1.

 

 

-O.8

 

Number of data points

FIGURE A.5

SAMPLE ﬂ
Eetimation of thermal conductivity

 

5583592179

>. 300 . _Serlest

100.

 

 

0 20 40 00 80 100 120
Number oi data pointa

FIGURE A.6

132

SAMELEi‘Z

 

 

 

SAMPLE #7
Measured vs Calculated

,..-..--_.-2_._2.--.- m . 45 _ 1

i

, 44 . K Q

d “

43.5 -

6 \ l

.. 43 - \ ;
g g + Series1
i .__.. Senesz

E 42.5 2 l

1-

. 1

/ 42 -2 1

1

1

1

I 41.5 - i

l

4 41 - ,

4a a i

-1 -0.6 -0.6 20.4 -0.2 0 0.2 0.4 0.8 0.6 1

rib

FIGURE A.7

133

SAMPLE #7
RESIDUALS

 

Temperature (C)
.6
N

04

~0.6

 

+ 5911031

 

FIGURE A.8

Number of data points

SAMPLE #7
Estimation of Thermal Conductivity

 

Thermal Conductivity (WIm K)

1003

 

FIGURE A.9

— Seriesi
688002872

‘0 60 w 100 120

Number oi data points

Temperature (C)

 

FIGURE A.10

-0.4 0.2

134

SAMPLE #3
Measured vs Calculated

.2 .._.2.-.H..M.__M‘

29.5 2 i 1

O

0.2 0.4 0.6 0.6 1

—e— $011031
+ $011082

135

SAMPLE #3
RESIDUALS

 

 

 

03 4»-

0.2 J»

0.1 ..

 

 

 

. .- . ( + 3011081
120 I”

 

 

 

 

20

Temperature (C)
o
3 1
1

 

-0.2 i)

 

 

 

 

Number of data points

FIGURE A.ll

SAMPLE 33

Etimetion of Thermal conductivity
1700
16m 4.
1500 .. .
1400 ..
1300 ..
1200 4.
1100 i

 

 

_ _— Series1
9143130833

Thermal Conductivity (Wlm K)

 

o§§§§§§§§

0 20 40 00 80 100 120 140
Number of data points

FIGURE A.12

Temperature (C)

 

Sample #2
Measured vs Calculated

106

A,

i“ ‘ ~.
4 tr

29.5 -

 

 

29 -
’4 28.5 -
\
I'\
28 .-
91 r.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.8 0.8

rib

FIGURE A.13

—e— $011031
—— Series2

l 3 7
RESIDUALS

SAMPLE #2

 

+ Series1
_ Senesi

 

 

 

 

 

 

0.6

-- - 2 2 1m: 1.... 211,121 .11-. i 1 1 :1 1---- 1: -1-.1-1.,.-...--..: 1 - 111 i- -
a
v
IN.
_ m
_ M...
'1
u
.6 m um
. 2 2m
m mm
m .m
E m m
_
a
'1
.ll‘ 1
4
J
A
4 n 0 . . L0 4 4 a 4 2i 4 a n n o 0 . n n
m mmmmm2mmmmmmmmmo
.0. taiEeh m 6. :55. £32.00 .252:

120 140

100

Number of data points

20

 

 

 

 

FIGURE A.15
SAMPLE #1
Measured vs Calculated
1? R 2
1 i
1 P I.
1 .
1 r ' Va 1
' 1
I 32 2 \\ 1
/ 1
1 \ 5
A 31.5 ~0 1
9' 1
a 1
:i l
E 1
E 31 - i
i
I 1
1
I 30.5 i
1
1 1
1
an i
-1 -0.8 -O.6 -04 .02 o 0.2 0.4 0.6 0.8 1
rib

FIGURE A.l6

+ $00031
— Senosz

139

SAMPLE I1
RESIDUALS

0.4

 

0.3
0.2 1

0.1 4.

J g + $011081

Residuals (C)

40 i 60 70 l 1 w

8
8

-O.2 .2

 

 

Number of data points

FIGURE A.l7

SAMPLE #1
Estimation of Thermal Conductivity

 

10m ..

1400 ..

1200 2.

947.”?2986

i
1

1M 1. — $011081

1

 

 

Thermal Conductiva (WIm K)

0001

400..

 

 

0 10 20 30 40 50 60 70 80 90
Number of data points

FIGURE A.l8

1
1

Temperature (C)

 

-1 -0.6

FIGURE A.l9

140

W

SAMPLE #19
Measured vs Calculated

46.

-O.2 0 0.2
rib

0.4

0.6

 

0.8

+ Series1
———— Series2

141

 

 

 

 

 

 

 

SAMPLE 819
RESIDUALS
1.5 -W... H MW-.- .3“... 22-....- ..
1 4
.. 0.5 .
9.
2
a
‘2’ + $611631
3
E
.2 o 2- - - _ j i
(l 10 20 30 0 50 60 70 r 90 100
-0.5 .. ‘;
-1
Number of data points
FIGURE A.20
SAMPLE #19
Estimation of Thermal Conductivity
600 - .3
500 1
.1?
E 400 ..
E
5:
5 ,
g 300 —Senes1
i \
O
3 2663514148
E 200 1
)5
100 .
0 .
O 10 20 30 40 50 60 70 80 90 100

Number of data points

142

 

 

 

 

FIGURE A.21
SAMBLEM
SAMPLE s4
Measured vs Calculated
r L5
. 6 .
.. '1.
V i
44.4 -
.. \
9.
2 44.2 . .
g ‘\ + $00081
—. Seriesz
a I’ 44 . " ‘ ' "
5 1 ‘h
43.6 - “
\
43.6 - \
1 1
43.4 - 5
ﬂ?
1 -o 6 o 6 0 4 .02 o 0 2 o 4 o 6 o 6 1
rib

FIGURE A.22

SAMPLE 84
RESIDUALS

0.3
0.2
0.1 ..

0, , _ ‘_ - . l -- .. .- ... g +Sen‘es1

Temperature (C)

-0.2 1.

-0.3 1

 

 

Number of data points

FIGURE A.23

SAMPLE #4
Estimation of Thermal Conductivity

 

— Seriosi

. 8826125331

L
f

 

Thermal Conductivity (Wlm K)
§

4
v ‘r

 

~42 ._ o... -..¢,*.,,...._......4-_

 

0 10 20 30 40 50 60 70 80 90 100
Number of dam points

FIGURE A.24

144

SAW

Sample #6
Measured vs Calculated

an)

.
‘
e

'2
2 + Seriesi
g 26 2 .. \
' i —— Series2
/ . 1
g 1
.. I 29 .. g
2/ :

 

1 . ..

5 i

1 , I

1 26.4 - ,

1 i
l

. 332 ‘1

FIGURE A.25

145

Sample #8
Residuals

 

Residuals (C)

 

 

-0.25

 

 

 

 

 

v

 

 

 

 

'

 

 

 

 

 

Number of data points

FIGURE A.26

1400

Sample '6
Estimation of Thermal Conductivity

 

1200.

.3

Thermal Conductivity (Wlm K)

 

7448016381

W

 

 

1 _ Seriosi

 

20 40 60 80 100 120
Number of data points

FIGURE A.27

146

APPENDIX B

FORTRAN PROGRAM NLIN

PROGRAM NLINA
CCCCCCCCC PROGRAM DESCRIPTION CCCCC
C C
C PROGRAM NLINC
C WRITTEN BY JAMES V. BECK C

C LAST REVISED JUL. 27, 1994 FOR ANDREW SAYERS

Cl##ttﬁttttttittt*lt*¢**#¥###ttl*ttttlitttllttttttttttttit##ttttttc

C C

CVCCCCCCCC VARIABLE IDENTIFICATION CCCCCCCC

C ' C

C C
Cteeeeeeeetseeeeteeesseseeeseseeeeesseeeeeassesses-seeeessssesesetec
C C

CDCCCCCCCC DIMENSION BLOCK BLOCK 0000

C C

C C

IMPLICIT REAL‘8 (A-H,O—Z)

DIMENSION T(IOOO,5),Y(1000),SIGZ(l000),B(5),Z(5),A(5),BS(5),
l VINV(5,5),BSS(5),CG(5),BSV(5),R(5,5),EXTRA(20),ERR( 1000)
I, PS(5,S),P(5,5),PSV(5,5),

I XTX(5,5),XTY(S),SUM(5),BET(50,2)

CHARACTER‘40 DFILE,OUTFIL
C C
Ceusesureeseesseeeeeeeesessseeeeeeeeeeeeueeueeeeeeeeueeseusuc
C C
COCCCCCCCC COMMON BLOCK BLOCK 0100
C C

COMMON T,N,Z,BS,I,ETA,PS,P,B,A,Y,SIG2,MODL,VINV,NP,EXTRA
COMMON/ERROR/ERR
COMMON/MOD/AA,TL,SUM,BET,IH,CONST

C C
C C
Cueseen-nuessen-asses"ueeeeseeeseeseueeeeeeseueeeeeenenesuc
C C
CACCCCCCCC DATA BLOCK BLOCK 0200
C C

DATA EPS,EPSS,IIN,IOUT/ I .0D-30,0.000 I D+0,5,7/
C C
Cursestsetse"-assessesareaeeseeeeeeeseeeeeueseeseeeusseesensneeec
C C
CICCCCCCCC INITIALIZATION BLOCK BLOCK 0400
C C

WRITE(‘,‘)'ENTER THE NAME OF THE DATA F ILE'
READ("','(A40)') DFILE

OPEN(8,FILE=DFILE)

WRITE(‘,")'ENTER THE NAME OF THE OUTPUT FILE'
READ("','(A40)') OUTFIL

OPEN(7,FILE=OUTFIL)

I47

geesetesstttestuanurnus"nun-scisun"*snnsssussuntumusc
C C

CPCCCCCCCC PROCESS BLOCK BLOCK 0500

C C

C --- START INPUT

C BLOCK I

WRITE(7,*)'BEGIN LISTING INPUT QUANTITIES
200 READ(8,"')N,NP,NT,ITMAX,MODL,IPRINT
WRITE(7,*)
WRITE(7,"‘)'BLOCK 1'
WRITE(7,*)'N = NO. DATA POINTS, NP = NO. PARAMETERS
WRITE(7,*)'NT = NO. OF INDEPENDENT VARIABLES'
WRITE(7,*)'ITMAX = MAXIMUM NO. OF ITERATIONS
WRITE(7,*)'MODEL = MODEL NUMBER, iF SEVERAL MODELS IN SUBROUTINES:
1 MODEL AND SENS‘ .
WRITE(7,*)'IPRINT = 1 FOR USUAL PRINTOUTS, 0 FOR LESS
WRITE(7,"')
IF(N.LE.0) THEN
STOP
END IF
WRITEC",'(/,9X,"N",8X,"NP",8X,"NT',5X,"ITMAX",5X,
+"MODEL",4X,"IPRIN'I"')')
WRITE(‘,'(7I 1 o)') N,NP,NT,ITMAX,MODL,IPRINT
WRITE(7,'(/,9X,"N",8X,"NP",8X,"NT',5X,"ITMAX",5X,
+"MODEL",4X,"IPRINT')')
WRITE(7,'(7I I 0y) N,NP,NT,ITMAX,MODL,IPRINT
IOPT=o
C -- IF IOPT=o THEN ON THE 2ND AND SUCCEEDING STACKED CASES, THE DATA IS
c -- NOT REPRINTED.
C IF IPRINT=I, EXTRA PRINT OUT OF ETA, RESIDUALS 30),... ARE GIVEN.
c BLOCK 2
WRITE(7,')
WRI'I’E(7,‘)'BLOCK 2'
WRITE(7,"')'B(1),B(2),..,B(NP) ARE INITIAL PARAMETER ESTIMATES
WRITE(7,*)
READ(8,‘)(B(I),I=I,NP)
WRITE(7,'(10X,"B(",ii,") = ",El6.5)') (I,B(I),I=I,NP)

DO 150 Jl=2,5
BS(Jl) = o
150 CONTINUE
C
IP(IOPT.LE.0) THEN
c BLOCK 3
WRITE(7,"')
WRITE(7,")’BLOCK 3'
WRITE(7,*)'J = DATA POINT INDEX, Y0) = MEASURED VALUE’
WRITE(7,*)'SIGMA(J) = STANDARD DEVIATION OF Y(J)'
WRITB(7,‘)'T(J, I) = FIRST INDEPENDENT VARIABLE’
WRITE(7,“)
WRITE(7,'(/,9X,"J",6X,"Y(J)",3X,"SlGMA(J)",6X,"T(J,i)"
+,6X,”T(J,2)")')
DO 10 12=1,N

148

READ(8,"‘)J,Y(J),SIG2(.I),(T(J,KT),KT= ] ,NT)
WRITE(7,'(I I 0,7F10.5)') J,Y(J),SIGZ(.I),(T(J,KT),KT: 1 ,NT)
SIG2(J) = SIGZ(J)*SIG2(J)
IO CONTINUE
END IF
C
313 D0 21P=I,NP
DO 2 KP=1,NP
PS(KP,IP) = 0
P(KP,IP) = O
CONTINUE
WRITE(7,'(/,5X,"P( l ,KP)",9X,"P(2,KP)",9X,"P(3,KP)",9X,
+"P(4,KP)",9X,"P(5,KP)")')
DO 6 IP=I,NP

READ(8,*)(PS(]P,KP),KP=1,NP)
WRITE(7,'(5D16.5)') (PS(IP,KP),KP= l ,NP)
CONTINUE
BLOCK 4
DO 88 IP=I,NP
88 PS(IP,IP)=B(IP)"B(IP)
READ(8,*)IEXTRA
C IEXTRA=0 FOR NO EXTRA INPUT WHICH COULD BE FOR CONSTANTS
C IN MODELS
C =1 FOR ONE INPUT, NAMELY: EXTRA(I), ETC.
WRITE(7,*)
WRITE(7,‘)'BLOCK 4' .
WRITE(7,‘)'IEXTRA = NO. OF EXTRAa) PARAMETERS, 0 IF NONE'
WRITE(7,*)
WRITE(7,'(10X,"IEXTRA = ",110)')IEXTRA
IF(IEXTRA .LT. I) GOTO 21
WRITE(7,"')
WRITE(7,’)'BLOCK 5'
WRITE(7,')'EXTRA(I),... ARE EXTRA CONSTANTS USED As DESIRED'
WRITE(7,"')
READ(8,‘)(EXTRA(1E),IE=| .IEXTRA)
WRITE(7,'("EXTRA(",12,") = ",F16.5)') (IE,EXTRA(IE),IE=I
I,IEXTRA)
21 CONTINUE
C
C -- ADD BLANK CARD AFTER LAST INPUT CARD
C u-END INPUT
WRITE(7,*)'END INPUT QUANTITIES - - BEGIN OUTPUT CALCULATIONS
WRITE(7,")
WRITE(7,")'SY = SUM OF SQUARES FOR PRESENT PARAMETER VALUES
WRITE(7,*)'SYP = SUM OF SQUARES FOR GAUSS PARAMETER VALUES, SHOULD
I BE SMALLER THAN SY’
WRITE(7,‘)' SYP DECREASES TOWARD A POSITIVE CONSTANT
WRITE(7,*)‘G = MEASURE OF THE SLOPE, SHOULD BECOME SMALLER AS
IITERATIONS PROCEED'
WRITE(7,*)' G SHOULD APPROACH ZERO AT CONVERGENCE
WRITE(7,‘)'H = FRACTION OF THE GAUSS STEP, AS GIVEN BY THE
lBOX-KANEMASU METHOD‘
WRITE(7,‘)

2
C
C
C
C
C
C
6
C

I49

WRITE(7,")
DO 18 IL=1,NP
BS(IL)=B(IL)
CG(IL) = o
18 CONTINUE
DO 19 [P=1,NP
XTY(IP)=0.0D+0
DO 19 KP=1,NP
P(KP,IP) = PS(KP,IP)
XTX(IP,KP)=0.0D+0
19 CONTINUE
1 = o
MAX=0
C
99 MAX = MAX + 1
c START BASIC LOOP GIVES 80) AND SY
C
SY = 0.0D+0
DO 100 13=1 ,N
1 = 13
CALL MODEL
CALL SENS
CCCC CALL MODEL
RISD = Y(I)—ETA
SY = SY + RISD’RISD/SIGZU)
SUMX= 0.0D+O
DO 20 K=1,NP
XTY(K)=XTY(K)+Z(K)*RISD/SIG2(I)
DO 20 L=1,NP
SUMX= SUMX+ Z(L)"'P(K,L)“Z(K)
XTX(K,L)= XTX(K,L) + Z(L)*Z(K)/SIGZ(I)
20 CONTINUE
DELTA = 5102(1) + SUMX
DO 29 II=1,NP
A(JJ) = com
29 CONTINUE
DO 30 JA=l,NP
DO 30 KA=1,NP
A(JA) = A(JA) + Z(KA)‘P(JA,KA)
3o CONTINUE
cs = 0.0D+0
DO 40 JC=1,NP
CS = cs + Z(JC)‘(B(JC)—BS(JC))
CG(JC) = CG(JC) + Z(JC)"‘RISD/SIGZ(I)
4o CONTINUE
C = Y(I) - cs - ETA
DO 50 IB=1,NP
B(IB) = B(IB) + (A(IB)’C)/DELTA
so CONTINUE
DO 41 ISV=1,NP
DO 41 JSV=I ,NP
PSV(ISV,ISV) = P(JSV,lSV)
41 CONTINUE
DO 52 IV=1,NP

150

D0 52 IU=lV,NP
SUMP = 0.0D+O
DO 51 KP=1,NP
DO 51 IP=I,NP
1F(I(P-lV.EQ.0.0R.JP-IU.EQ.O) GOTO 51
PSQ] = PSV(KP,JP)*PSV(IU,1V)
PSQZ = PSV(1U,KP)*PSV(1V,JP)
PSQ = PSQ] - PSQZ
IF(DABS(PSQ])+DABS(PSQZ).LT.1.D-15) TH EN
RP = PSQ * 1.DlS
ELSE
RP = PSQ / (DABS(PSQI)+DABS(PSQZ))
END IF
RP = ABS(RP)
RPP = RP -1.0D-12
IF(RPP.LE.0.0D+O) THEN
PSQ = 0.0D+0
END IF
SUMP = SUMP + Z(JP)"‘Z(KP)‘PSQ
51 CONTINUE
P(IU,IV) = (PSV(IU,IV)"‘SIGZ(I)+SUMP)/DELTA
52 CONTINUE
DO 53 IV=2,NP
IVM = IV - 1
DO 53 [U = I,IVM
P(IU,IV)= P(IV,IU)
53 CONTINUE
IF(IPRINT.GT.0) THEN
IF(I.EQ.1) THEN
WRITE(7,"')
WRITE(7,")’SEQUENTIAL ESTIMATES OF THE PARAMETERS GIVEN BELOW'
WRITE(7,'(//,3X,"1",6X,"ETA",3X,"RES.",2X,
1"B( l )",7X,"B(2)",7X,"B(3)",7X,"B(4)")')
END IF
C WRITE(7,'(I4,6El2.5)')I,ETA,RISD,(B(JC),JC=I,NP)
WRITE(7,'(I3,F10.2,F8.3,5E11.4)')i,ETA,RISD,(B(JC),JC=I,NP)
END [F
100 CONTINUE
C --- END BASIC LOOP, GIVES 80) AND SY
C -- START BOX-KANEMASU MODIFICATION
C
C START BOX-KANEMASU MODIFICATION
IF(MAX-I)104,104,103
103 SS=SY/2.0D+O
IF(SS-SYP)104,104,105
105 D0 210 IBS=I,NP
B(IBS)= BSV(IBS)
210 CONTINUE
WRITE(IOUT,212)
212 FORMAT(7X,'USE BSV(IBS)')
GOTO 21 1
104 CONTINUE
DO 102 IBS=I,NP
BSS(IBS)= BS(IBS)

IS]

102 CONTINUE
ALPHA= 2.0D+0
AA= 1.1D+0
110 ALPHA= ALPHA/2.0D+O
DO 116 IBS=I,NP
BS(IBS)= BSS(IBS) + ALPHA*( B(IBS)-BSS(IBS) )
BSV(IBS)= BS(IBS)
116 CONTINUE
INDEX=O
G= 0.0D+O
D01151P=1,NP
DELB= BS(IP)-BSS(IP)
G= G + DELB‘CGGP)
RATIO= DELB/( BSS(IP)+EPS )
RATIO= ABS(RATIO)
IF(RATIO-EPSS)I 13,1 13,114
113 INDEX= INDEX+1
WRITE(IOUT,314)
314 FORMAT(7X,'MAX',8X,'NP',5X,'INDEX',8X,'lP')
WRITE(7,'(7110)') MAX,NP,INDEX,IP
114 CONTINUE
C WRITE(7,122) 1,Y(I),ETA,RISD,Z(1P),XYP,DELB,SIG2(I)
115 CONTINUE
SYP= 0.0D+0
DO 117 I3=1,N
I=I3
CALL MODEL
RISD= Y(I)«ETA
SYP= SYP + RISD‘RISD/SIGZO)
117 CONTINUE
IF(NP-INDEX)106,106,IO7
106 H=1.0D+0
GOTO 132
107 CONTINUE
SYN= SYP‘0.999D+0
IF(SYN-SY)1 12,1 12,111
11 1 IF(ALPHA-0.01D+0)109,109,l10
109 WRITE(7,108)ALPHA,SYP,SY
108 FORMAT(3X,'ALPHA TOO SMALL,ALPHA=’,F12.6,2X,'SYP=',E15.6,2X,
1'SY',E15.6)
WRITE(7,1001)
1001 FORMAT(8X,'Z(1)',10X,'Z(2)',10X,'Z(3)',10X,'Z(4)',10X,'Z(5)')
1002 FORMAT(6E13.4)
DO 1003 I=1,N
CALL SENS
WRITE(7,1002) (Z(IBB),IBB=1,NP)
1003 CONTINUE
GOTO 1000
112 CONTINUE
SKSUM= SY - ALPHA‘G‘( 2.0D+0-1.0D+0/AA )
IF(SYP-SKSUM)131,131,130
130 H= ALPHA * ALPHA’G/( SYP-SY+2.0D+0*ALPHA*G )
GOTO 132
131 CONTINUE

152

H= ALPHA*AA

132 CONTINUE
DO 118 IBN= 1,NP
B(IBN)= BSS(IBN) + H * ( B(IBN)-BSS(IBN) )

118 CONTINUE

211 CONTINUE
WRITE(IOUT,121)

WRITE(*,121)

121 FORMAT(5X,‘MAX',IOX,'H',I3X,'G',12X,
1'SY',1 1X,'SYP')

WRITE(7,122) MAX,H,G,SY,SYP
WRITE(*,122) MAX,H,G,SY,SYP

122 FORMAT(I8,IF13.6,4E14.6)
WRITE(7,’(10X,"B(",I1,") = ",E16.6)')(I,B(1),I=1,NP)
WRITE(*,'(10X,"B(",11,") = ",E16.6)‘) (I,B(I),i=1,NP)

C END BOX-KANEMASU MODIFICATION
WRITE(7,'(/,SX,"P(l,KP)",9X,"P(2,KP)",9X,"P(3,KP)",9X,
1"P(4,KP)",9X,"P(5,KP)")')

DO 206 IP=I,NP
WRITE(7,207) (P(IP,KP),KP=1,NP)
206 CONTINUE
207 FORMAT(5D15.7)
WRITE(7,135)

135 FORMAT(5X,'CORRELATION MATRIX')
DO 136IR=1,NP
DO 136 IR2=1,IR

AR= P(IR,IR) * P(IR2,IR2)
R(IR,IR2)= P(IR,IR2)/SQRT(AR)

136 CONTINUE
DO 137IR=1,NP
WRITE(7,'(5E15.7)') (R(IR,III),III=1,IR)

137 CONTINUE
DO 126 1PS=1,NP
PS(IPS,IPS)= (1.0E+7) . P(IPS,IPS)

126 CONTINUE
WRITE(7,’)'XTX(I,K),K=1,NP'

DO 220 K=1,NP

220 WRITE(7,'(5E15.7)')(XTX(K,111),III=1,NP)
WRITE(7,‘)'XTY(I),I=1,NP, WHERE Y [S RESID’
WRITE(7,'(5E1S.7)')(XTY(1),I=1,NP)

127 FORMAT(3X,'IPS=',I4,3X,'PS(IPS,IPS)=',D15.8)
WRITE(7,‘)'XTY(I),I=1,NP, Y IS Y, NOT RESID'
WRITE(7,'(5E15.7)')(SUM(I),I=1,NP)

DO 119IP=1,NP
XTY(IP)=0.0D+O
DO 119 KP=1,NP
P(IP,KP)= PS(IP,KP)
XTX(IP,KP)=0.0D+O

119 CONTINUE
DO 1201P=1,NP
BS(IP)= B(IP)

CG(IP)= 0.0D+0

120 CONTINUE

WRITE(7,314)

153

WRITE(7,'(7IIO,4F10.4)') MAX,NP,INDEX,IP
IF(NP-INDEX)IOI,IOI,123
123 CONTTNUE
M=ITMAX
IF (MAX-M)99,99, I 0 I
I0] CONTINUE
IF(IPRINT)I33,I33,134
133 IPRINT=IPRINT+I

GOTO 99
134 CONTINUE
C
1000 CONTINUE
CLOSE(IIN)
CLOSE(IOUT)
c c
Citiﬁtiltttﬁllﬁttttt*tttlﬁtltititttiiltttt##tttltttttttﬁtttttttttic
c C
CECCCCCCCC ERROR MESSAGES BLOCK 0900
C C
C C
Ct!$.0¢*3*#‘##¥#*$#**¥*00ittﬁtitﬁltiﬁitﬁtt$.00!Iiﬁttttlﬁﬁttﬁitt$$ﬁc
C C
CFCCCCCCCC FORMAT STATEMENTS BLOCK 9000
C C
c c

Clttiﬁti##t#*#¥.#*¥tti.ttIt.ittt0*tittttﬁttﬁtﬁtittttlﬁttttitttﬁtﬁtc

C
STOP
END
SUBROUTINE MODEL

C THIS SUBROUTINE IS FOR CALCULATING ETA, THE MODEL VALUE
IMPLICIT REAL‘8 (A-H,O-Z)
DIMENSION T(1000,5),Y(1000),SIGZ(1000),B(5),Z(5),BET(50,2),
+A(5),BS(S),VINV(S,5),EXTRA(20)
DIMENSION P(5,5),PS(5,5),SUM(5)
COMMON T,N,z,Bs,1,ETA,Ps,P,B,A,Y,SIoz,MODL,VINV,NP,EXTRA
COMMON/MOD/AA,TL,SUM,BET,IH,CONST

C WRITTEN BY JAMES v. BECK
PI=4.0D+O"‘DATAN(1.0D+0)

c IF(MODL .EQ. 1) GOTO 800

800 CONTINUE

TR=T(I,1 )/EXTRA(2)
ETA=BS( l )+BS(2)‘ EXTRA( I )‘( I .OD+O-TR"TR)

1000 CONTINUE
C WRITE(‘,’)'1,T(I,1),ETA,Z(I)',I,T(1,l),ETA,Z(1)
RETURN
END
SUBROUTINE SENS
C THIS SUBROUTINE IS FOR CALCULATING THE SENSITIVITY COEFFICIENTS
IMPLICIT REAL‘B (A-H,O-Z)
DIMENSION T(1000,5),Y(1000),SIG2(1000),B(5),BET(50,2),
+Z(5),A(5),BS(5),VINV(5,5),EXTRA(20)

154

DIMENSION P(5,5),PS(5,5),SUM(5)

COMMON T,N,Z,BS,I,ETA,PS,P,B,A,Y,SIGZ,MODL,VINV,NP,EXTRA
COMMON/MOD/AA,TL,SUM,BET,IH,CONST
PI=4.0D+O*DATAN(1.0D+O)

Z( 1 )=1 .OD+O
TR=T(I,1 )/EXTRA(2)
Z(2)=EXTRA(1)*(1.0D+0-TR*TR)

DO 312 1PP=1,NP
312 SUM(IPP)=0.0D+0
313 CONTINUE
800 CONTINUE
C IF(I .LT. N)GOTO 2000
DO 1001 JPP=1,NP
C TZ=? TRY ETA FOR NOW
Tz=ETA
1001 SUM(IPP)=SUM(JPP)+Z(JPP)*(Y(1)-TZ)/SIGZ(I)
2000 CONTINUE
RETURN
END

155

ABEENDIXL:
NLIN OUTPUT AND INPUT FILES

Sample #4
Output file

BEGIN LISTING INPUT QUANTITIES

BLOCK 1

N = NO. DATA POINTS, NP = NO. PARAMETERS

NT = NO. OF INDEPENDENT VARIABLES

ITMAX = MAXIMUM NO. OF ITERATIONS

MODEL = MODEL NUMBER, IF SEVERAL MODELS IN SUBROUTINES: MODEL AND SENS
IPRINT = I FOR USUAL PRINTOUTS, 0 FOR LESS

N NP NT ITMAX MODEL IPRINT
93 2 1 100 I I

BLOCK 2
B(I),B(2),..,B(NP) ARE INITIAL PARAMETER ESTIMATES

B(I) = .43500E+02
B(2) = .25000E-02

BLOCK 3

J = DATA POINT INDEX, Y(J) = MEASURED VALUE
SIGMA(J) = STANDARD DEVIATION OF Y(J)

T(J,l) = FIRST INDEPENDENT VARIABLE

1 YO) SIGMA(J) T(.l,1) T(J,2)
1 43.65344 1.00000 4.00000
2 43.55461 1.00000 -.97826
3 43.65344 1.00000 -.95652
4 43.60403 1.00000 -.93478
5 43.65344 1.00000 -.91304
6 43.75226 1.00000 -.89130
7 43.75226 1.00000 -.86957
8 43.90050 1.00000 -.84783
9 43.85109 1.00000 -.82609
10 43.94992 1.00000 -.80435
11 43.99933 1.00000 -.78261
12 43.94992 1.00000 -.76087
13 44.09816 1.00000 -.73913
14 44.04874 1.00000 -.71739
15 44.09816 1.00000 -.69565
16 44.04874 1.00000 -.67391
17 44.04874 1.00000 -.65217
18 44.14757 1.00000 -.63043
19 44.09816 1.00000 -.60870

20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
54
65
66
67
68
69
70
71
72
73

44.34522
4439464
49444405
44.49346
4444405
44.59229
44.64170
44.74053
44.74053
44.59229
44.64170
¢¥L64l70
44.691 1 1
44.74053
44.691 1 1
44.83935
44.74053
44.83935
44.83935
44.83935
44.93818
44.83935
44.83935
44.78994
44.83935
44.93818
44.83935
44.83935
44.74053
44.83935
4483935
44.69] 11
44.74053
40464170
44.74053
44.74053
44.74053
44.83935
44.74053
44.78994
44.74053
44.74053
44.78994
lh464l70
44.691 1 1
49464170
44.59229
44.59229
4444405
44.49346
4444405
4444405
44.49346
44.34522

LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
L00000
LOOOOO
LOOOOO
L00000
L00000
L00000
L00000
L00000
L00000
LOOOOO
LOOOOO
L00000
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
LOOOOO
L00000
L00000
L00000
L00000
LOOOOO
L00000
L00000
L00000
L00000
L00000
L00000
L00000
LOOOOO
LOOOOO
L00000
L00000
L00000
L00000
L00000
L00000
L00000
L00000
LOOOOO
L00000

«58696
«56522
«54348
«52174
«50000
«47826
«45652
«43478
«41304
«39130
«36957
«34783
«32609
«30435
«28261
«26087
«23913
«21739
«19565
«17391
«15217
«13043
«10870
«08696
«06522
«04348
«02174
.00000
.02174
.04348
.06522
.08696
.10870
.13044
.15217
.17391
.19565
.21739
.23913
.26087
.28261
.30435
.32609
.34783
.36957
.39130
.41304
.43478
.45652
.47826
.50000
.52174
.54348
.56522

156

74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93

44.39464
44.24640
44. 14757
44.24640
44.19698
44.29581
44.24640
44.24640
44.24640
44.19698
44.24640
44.09816
43.99933
43.94992
43.80168
43.85109
43.75226
43.70285
43.70285
43.55461

BLOCK 4
IEXTRA = NO. OF EXTRA(I) PARAMETERS, 0 IF NONE

IEXTRA =

BLOCK 5
EXTRA(I),... ARE EXTRA CONSTANTS USED AS DESIRED

EXTRA( 1) =
EXTRA( 2) =
END INPUT QUANTITIES - - BEGIN OUTPUT CALCULATIONS

LOOOOO
LOOOOO
LOOOOO
LOOOOO
L00000
L00000
LOOOOO
L00000
L00000
LOOOOO
LOOOOO
L00000
LOOOOO
L00000
L00000
LOOOOO
LOOOOO
LOOOOO
L00000
L00000

2

1228.90000

1.00000

.58696
.60870
.63044
.65217
.67391
.69565
.71739
.73913
.76087
.78261
.80435
.82609
.84783
.86957
.89130
.91304
.93478
.95652
.97826
L00000

157

SY = SUM OF SQUARES FOR PRESENT PARAMETER VALUES
SYP = SUM OF SQUARES FOR GAUSS PARAMETER VALUES, SHOULD BE SMALLER THAN

SY

SYP DECREASES TOWARD A POSITIVE CONSTANT

G = MEASURE OF THE SLOPE, SHOULD BECOME SMALLER AS ITERATIONS PROCEED
G SHOULD APPROACH ZERO AT CONVERGENCE

H = FRACTION OF THE GAUSS STEP, AS GIVEN BY THE BOX-KANEMASU METHOD

SEQUENTIAL ESTIMATES OF THE PARAMETERS GIVEN BELOW

1 ETA RES. B(I)
43.50 .153
43.63 «078
43.76 «108
43.89 «284
44.01 «358
44.13 «379
44.25 «497

B(2)
.4365 E+02
.43 54E+02
.4349E+02
.4344E+02
.4340E+02
.4338E+02
.4337E+02

B(3)
.2500E-02
.2462E-02
.24 1 7E-02
.2300E-O2
.2 162 E-02
20445-02
. I 889E-02

3(4)

\IGUDAUJN—

8 44.36 «463 .4337E+02 .1809E-02
9 44.48 «625 .4337E+02 .1668E-02

10
l 1
12
13
I4
15
16
17
18
I9
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61

44.58
44.69
44.79
44.89
44.99
45.09
45.18
45.27
45.35
45.43
45.51
45.59
45.66
45.74
45.80
45.87
45.93
45.99
46.05
46.10
46.15
46.20
46.25
46.29
46.33
46.36
46.40
46.43
46.45
46.48
46.50
46.52
46.54
46.55
46.56
46.57
46.57
46.57
46.57
46.57
46.56
46.55
46.54
46.52
46.50
46.48
46.45
46.43
46.40
46.36
46.33
46.29

«635
«691
«844
«796
«942
«987
-1.128
-1.217
-1.204
-1.336
-1.169
-1.196
- I .221
-1.242
-l.360
-I .277
-1.290
-1.251
-I.308
-l.510
-1.511
-1.559
-1.554
-1.547
-1.636
-I.524
-1.656
-1.588
-I .615
-1.640
-I.563
-1.681
-1.697
-1.759
-1.720
-1.628
-1.731
-1.733
-1.830
-1.727
-1.720
-I .858
-I.795
-I.878
-l.761
-I.739
-1.714
-I.588
-I.656
-I.573
-l.586
-l.547

.4338E+02
.4339E+02
.4340E+02
.4340E+02
.4341E+02
.4342E+02

.4344E+02
.4345E+02
.4346E+02
.4347E+02
.4347E+02
.4346E+02
.4346E+02
.4345E+02
.4345E+02
.4344E+02
.4343E+02
.4342E+02
.4342E+02
.4342E+02
.4342E+02
.434ZE+02
.4342E+02
.4342E+02
.4342E+02
.4342E+02
.4342E+02
.4342E+02
.4342E+02
.434ZE+02
.434IE+02
.434lE+02
.4341E+02
.4341E+02
.4342E+02
.4341E+02
.4341E+02
.4341E+02
.434ZE+02
.4342E+02
.4342E+02
.4342E+02
.4342E+02
.4343E+02
.4343E+02
.4343E+02
.4343E+02
.4343E+02
.434BE+02
.4343E+02
.4343E+02
.4343E+02

.1580E-02
.1506E-02
.1394E-02
.1358E-02
. 1285 E-02
.1232E-02
.1 158E-02
.1087E-02
.1053E-02
. 1001 E-02
.1020E-02
.1039E-02
.1059E-02
.1079E-02
.1079E-02
.I 102E-02
.1 124E-02
.1 156E-02
.1 1785-02
.1 170E-02
.I 167E-02
.1 162E-02
.1 161E-02
.I I64E-02
.1 159E-02
.1 I68E-02
.l 165E-02
.1 l70E-02
.1 I74E-02
.1 I76E-02
.1 185E-02
.1 185E-02
.I I84E-02
.1 179E-02
.1 178E-02
.I 183E-02
.1 18 I E-02
.1 I80E-02
.I 173 E-02
.l I72E-02
.I 171E-02
.1 163E-02
.1 159E-02
.1 150E-02
.1 147E-02
.1 I45E-02
.1 l44E-02
.1 l46E-02
.1 I45E-02
.1 147E-02
.1 147E-02
.1 148E-02

158

159

62 46.25 -1.456 .4343E+02 .1151E-02
63 46.20 —1.559 .4343E+02 .1150E-02
64 46.15 -I.462 .4343E+02 .1151E-02
65 46.10 -I.460 .4343E+02 .1151E-02
66 46.05 -1.456 .4343E+02 .1151E-02
67 45.99 -1.399 .4343E+02 .1151E-02
68 45.93 -1.488 .4343E+02 .1150E-02
69 45.87 -I.376 .4343E+02 .1150E-02
70 45.80 -1.360 .4343E+02 .1150E-02
71 45.74 -1.292 .4343E+02 .1150E-02
72 45.66 -1.171 .4343E+02 .1150E-02
73 45.59 -I.246 .4343E+02 .1150E-02
74 45.51 -1.119 .4343E+02 .1150E-02
75 45.43 -I.188 .4343E+02 .1151E-02
76 45.35 -1.204 .4342E+02 .1153E-02
77 45.27 -1.019 .4342E+02 .1153E-02
78 45.18 «980 .4342E+02 .1153E-02
79 45.09 «790 .4343E+02 .1149E-02
80 44.99 «745 .4343E+02 .1144E-02
81 44.89 «647 .4344E+02 .1138E-02
82 44.79 «547 .4345E+02 .1129E-02
83 44.69 «494 .4346E+02 .1120E-02
84 44.58 «338 .4348E+02 .1106E-02
85 44.48 «378 .4349E+02 .1096E-02
86 44.36 «365 .4350E+02 .1089E-02
87 44.25 «299 .4350E+02 .1082E-02
88 44.13 «330 .4351E+02 .1081E-02
89 44.01 «160 .4351E+02 .1074E-02
90 43.89 «135 .4352E+02 .1069E-02
91 43.76 «058 .4352E+02 .1065E-02
92 43.63 .071 .4353E+02 .1057E-02
93 43.50 .055 .4353E+02 .1056E-02
MAX H G SY SYP
1 1.002257 .I48214E+03 .149325E+03 .776581E+00
B(I)= .435313E+02
B(2)= .105239E-02
P( 1 ,KP) P(2,KP) P(3,KP) P(4,KP) P(5,KP)

.6054330D-01 «6144288D-04

«6144288D-04 .7582198D—07
CORRELATION MATRIX

.1000000E+01

«9068615E+00 .1000000E+01
XTX(1,K),K=1 ,NP

.9300000E+02 .7536364E+05

.7536364E+05 .7410024E+08
XTY(I),I=I,NP, WHERE Y 18 RESID
« 1059492E+03 «1049059E+06
XTY(I),I=I,NP, Y IS Y, NOT RESID

.5461000E-01 .0000000E+00

MAX NP INDEX IP
I 2 0 3

SEQUENTIAL ESTIMATES OF THE PARAMETERS GIVEN BELOW

.—

\OWQO‘MhWNﬂ

ETA
43.53
43.59
43.64
43.69
43.75
43.80
43.85
43.89
43.94

43.99

44.03

44.08

44.12

44.16

44.20

44.24

44.27

44.31

44.35

44.38

44.41

44.44

44.47

44.50

44.53

44.56

44.58

44.60

44.63

44.65

44.67

44.69

44.70

44.72

44.74

44.75

44.76

44.78

44.79

44.79

44.80

44.81

44.81

44.82

44.82

44.82

44.82

44.82

44.82

44.82

44.81

RES.
.122
«032
.012
«090
«093
«045
«094
.006
«091
«038
«033
«126
«020
«1 10
-. 101
«I88
«226
«163
«247
«034
«017
.001
.021
«057

.087
.160
.137
«034
«006
«026
.004
.036
«030
.103
«010
.076

.054
.144
.037
.030
«025
.020
. l 16
.015
.015
«083
.017
.020
«124

3(1)

.4365E+02

B(2)
. 1052E-02

B(3)

.4365E+02 «1867E-02

.4362E+02 -
.4362E+02 -

.4361E+02
.4359E+02
.4359E+02

.4356E+02

.4356E+02
.4355E+02
.4354E+02
.4355E+02
.4354E+02
.4354E+02
.4354E+02
.4355E+02
.4356E+02
.4356E+02
.4356E+02
.4355E+02
.4354E+02
.4353E+02
.4352E+02
.4351E+02
.4350E+02
.4349E+02
.4347E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4345E+02

.6746E-05
.9759E-04
.9343 E-04
.4122E-03
.4887E-03
.7289E-03
.7356E-03

.812 I E-03
.869OE-03
.836 l E-03
.893 1 E-03
.8805E-03
.8804E-03
.8426E-03
.8035E-03
.8017E-03
.7747E-03
.8228E-03
.8664E-03
.9064E-03
.9435E-03
.9565E-03
.9923E-03
.1026E-02
.1067E-02
.1097E-02
.1095E-02
. IO98E-02
. 1097E-02
.1 100E-02
. I 106E-02
. l 104E-02
.1 1 17E-02
.1 I 16E-02
.l 124E-02
. 1 I30E-02
. 1 134E-02
.1 l45E-02
.1 146E-02
. l I47E-02
.1 143E-02
.1 143E-02
.I 149E-02
. l 149E-02
.1 l48E-O2
. I 141 E-02
. I 141 E-02
. I 141E-02
. 1 134E-02

B(4)

160

52 44.81 «069
53 44.80 «161
54 44.79 -.054
55 44.79 -.045
56 44.78 -.035
57 44.76 .076
58 44.75 -010
59 44.74 .053
60 44.72 .019
61 44.70 .036
62 44.69 .103
63 44.67 «026
64 44.65 .043
65 44.63 .015
66 44.60 -.012
67 44.58 .012
68 44.56 -.111
69 44.53 -.035
70 44.50 -.057
71 44.47 «028
72 44.44 .051
73 44.41 «066
74 44.38 .016
75 44.35 -099
76 44.31 «163
77 44.27 «028
78 44.24 -.040
79 44.20 .097
80 44.16 .087
81 44.12 .128
82 44.08 .171
83 44.03 .165
84 43.99 .259
85 43.94 .156
86 43.89 .104
87 43.85 .103
88 43.80 .005
89 43.75 .105
90 43.69 .058
91 43.64 .062
92 43.59 .116
93 43.53 .023
MAX H
2 1.000000

B(I)=

B(2)=

.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4346E+02
.4345E+02
.4345E+02
.434SE+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.434SE+02
.4346E+02
.4346E+02
.4347E+02
.4348E+02
.4350E+02
.4351E+02
.4352E+02
.4352E+02
.4352E+02
.4353E+02
.4353E+02
.4354E+02
.4354E+02
.4355E+02

G

.1 129E-02
.1 12 1 E-02
.1 I 19E-02
.I 1 17E-02
.1 1 16E-02
.1 1 18E-02
.l I 18E-02
. 1 120E-02
.1 120E-02
.1 121E-02
.1 124E-02
.I 123E-02
.1 124E-02
.1 125E-02
.1 125E-02
.1 125E-02
.1 124E-02
.1 124E-02
.1 l24E-02
.1 124E-02
.1 124E-02
.1 124E-02
.1 123E-02
.I 125E-02
.1 127E-02
.1 1275-02
.1 127E-02
.I 123E-02
.l I 19E-02
.I I 12E-02
.1 104E-02
.1095E-02
.1081E-02
.1072E-02
.1066E-02
.1060E-02
.1059E-02
.1052E-02
.1049E-02
.1045E-02
.1039E-02
.1038E-02

SY

161

.334621E-02 .775828E+00 .772482E+00
.435456E+02
.103792E-02

P(1,KP) P(2,KP) P(3.KP)
.61 15672D-01 «6219943D-04
«6219943D-04 .7675514D—07
CORRELATION MATRIX
.1000000E+01
«9078426E+00 .1000000E+01
XTX(I,K),K=1 ,NP

P(4,KP) P(5,KP)

162

.9300000E+02 .7536364E+05

.7536364E+05 .7410024E+08
XTY(I),I=I,NP, WHERE Y IS RESID

.2391551E+00 .5166700E+01
XTY(I),I=1 ,NP, Y IS Y, NOT RESID
.2333597E-01 .0000000E+00

MAX NP INDEX
2 2 0 3

IP

SEQUENTIAL ESTIMATES OF THE PARAMETERS GIVEN BELOW

ETA RES. B( I) B(2) B(3) 3(4)

--OOOOQOsMwa—Oxoooqoxugum.—°\oggqa‘u.&wN—°\OOO\IO\Mth-—-._

43.55
43.60
43.65
43.71
43.76
43.81
43.86
43.90
43.95
44.00
44.04
44.08
44.12
44.16
44.20
44.24
44.28
44.31
44.35
44.38
44.41
44.44
44.47
44.50
44.53
44.56
44.58
44.60
44.63
44.65
44.67
44.69
44.70
44.72
44.73
44.75
44.76
44.77
44.78
44.79
44.80

.108
«046
«001
«102
«104
«056
«104
«004
«100

«046

«041

« I 33

«026

«1 16

«106

«193

«230

«167

«250

«036

«019

.000
.020
«058
.063
.086
.161
.137

«033

«005

«025

.006
.038
«028
.105
«008
.079
.067
.057
.147
.040

.4365E+02 .1038E-02
.4365E+02 « I 867E-02
.4362E+02 «6752E-05
.4362E+02 «9760E-04

.4361E+02 .
.4359E+02 .
.4359E+02 .
.4356E+02 .
.4356E+02 .

.4355E+02
.4354E+02
.4355E+02
.4354E+02
.4354E+02
.4354E+02
.4355E+02
.4356E+02
.4356E+02
.4356E+02
.4355E+02
.4354E+02
.4353E+02
.4352E+02
.4351E+02
.4350E+02
.4349E+02
.4347E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4346E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4344E+02
.4344E+02

9343 BM

4 122E-03

4887E-03

7289E-03

73 56E-03

.8 121 E-03
.8690E-03
.8361 E-03
.893 1 E-03
.8805E-03
.8804E-03
.8426E-03
.8035 E-03
.80 1 7E-03
.7747E-03
.8228E-03
.8664E-03
.9064E-03
.9435E-03
.9565E-03
.9923 E-03
. 1026E-02
. 1067E-02
. 1097E-02
. 1095E-02
. 1098E-02
. 1097E-02
. 1 100E-02
. 1 106E-02
. I 104E-02
.1 1 17E-02
. 1 1 16E-02
. l 124E-02
. 1 130E-02
. l 134E-02
. 1 I45E—02
. 1 l46E-02

42
43

45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63

65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83

85
86
87
88
89

91
92
93

44.81
44.81
44.82
44.82
44.82
44.82
44.82
44.82
44.82
44.81
44.81
44.80
44.79
44.78
44.77
44.76
44.75
44.73
44.72
44.70
44.69
44.67
44.65
44.63
44.60
44.58
44.56
44.53
44.50
44.47
44.44
44.41
44.38
44.35
44.31
44.28
44.24
44.20
44.16
44. 12
44.08
44.04
44.00
43.95
43.90
43.86
43.81
43.76
43.71
43.65
43.60
43.55
MAX
3

.033
«021
.024
.120
.019
.018
«080
.021
.024
«120
«065
«158
«051
«042
«032
.079
«008
.056
.021
.038
.104
«025
.044
.016
«01 1
.012
«1 1 1
«036
«058
«030
.049
«068
.013
«102
«167
«032
«045
.092
.082
.122
.164
.157
.25 1
.148
.095
.093
«006
.093
.046
.049
.102
.009
NP

2

.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4344E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4346E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.434SE+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4345E+02
.4346E+02
.4346E+02
.4347E+02
.4348E+02
.4350E+02
.4351E+02
.4352E+02
.4352E+02
.43525+02
.4353E+02
.4353E+02
.4354E+02
.4354E+02
.4355E+02
HVEHEX

I I

. 1 147E-02
. 1 I43E-02
. 1 143E-02
. 1 149E-02
. I 149E-02
. 1 148E-02
. 1 14 1 E-02
. I 141 E-02
.1 141E-02
. I 134E-02
. 1 129E-02
. 1 121E-02
.1 l 19E-02
.1 1 17E-02
.1 I 16E-02
.I 1 18E-02
.1 1 18E-02
.1 120E-02
.1 120E-02
.1 121E-02
.1 1245-02
.1 123E-02
.1 124E-02
.1 125E-02
. 1 125E-02
.I 125E-02
.1 124E-02
.1 124E-02
.1 124E-02
.I 124E-02
.1 124E-02
.1 I24E-02
.1 123E-02
.1 125E-02
.1 127E-02
.1 127E-02
.1 127E-02
.1 123E-02
.l 1 19E-02
.1 I 12E-02
.1 104E-02
.1095E-02
.1081E-02
.1072E-02
.1066E-02
.106OE-02
.1059E-02
.1052E-02
.1049E-02
.1045E-02
.1039E-02
.1038E-02

IP

163

MAX NP INDEX
3 2 2 2
MAX H G SY
3
B(1) = .435456E+02
B(2) = .103792E-02
P( 1 ,KP) P(2,KP)

IP

164

SYP

1.000000 .542254E-17 .772482E+OO .772482E+00

P(3,KP)

.61 15672D-OI «6219943D-04
«6219943D-04 .7675514D—07
CORRELATION MATRIX

. 1000000E+0 1
«9078426E+00

XTX(1,K),K=1 ,NP
.9300000E+02 .7536364EZ+05
.7536364E+05 .7410024E+08

XTY(I),I=I,NP, WHERE Y IS RESID

. 1000000E+0 1

«2028688E-07 «2004377E-04

XTY(I),I=I,NP, Y IS Y, NOT RESID

.903 13878-02 .0000000E+00

MAX NP INDEX
3 2 2 3

105 2 1 100 l 1
26 0.0025
1 27.77786 1 -l
2 28.05399 1 -0.98077
3 28.19208 1 -0.96154
4 28.39914 1 -0.9423 1
5 28.67529 1 -0.92308
6 28.95142 1 -0.90385
7 29.0882 1 -0.88462
8 29.22266 1 -O.86538
9 29.28989 1 -0.84615
10 29.42438 1 -0.82692
11 29.35712 1 -0.80769
12 29.62604 1 -0.78846
13 29.55881 1 -0.76923
14 29.62604 1 -0.75
15 29.76053 1 -0.73077
16 29.62604 I -0.71 154
17 29.55881 1 0.69231
18 29.89496 1 -0.67308
19 30.29837 1 -0.65385
20 30.70175 1 -0.63462
21 30.90347 1 -0.61538
22 30.83621 1 -0.59615
23 30.63452 1 -0.57692
24 30.50006 1 -0.55769
25 30.50006 1 -0.53846
26 30.70175 1 -0.51923

1P

P(4,KP)

P(5,KP)

Input ﬁle

27
28
29
3O
31
32
33
34
35
36
37
38
39
40
41
42
43

45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63

65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80

30.83621
30.9707

31.10516
31.23963
31.23963
31.37409
31.37409
31.10516
31.37409
31.50852
31.57578
31.64191
31.64191
31.83847
31.70743
31.57578
31.90399
31.83847
31.96951
31.90399
32.03504
31.90399
31.83847
31.90399
32.03504
32.23163
32.43912
32.16608
32.29715
32.03504
32.03504
31.90399
31.96951
32.03504
32.29715
32.23163
32.03504
31.90399
32.03504
32.03504
31.96951
31.83847
31.77295
31.77295
31.50852
31.44129
31.17236
31.30686
31.30686
31.23963
31.17236
31.10516
31.03793
31.10516

1

-0.5
-0.48077
-0.46154
-O.4423 1
-O.42308
-0.40385
-O.3 8462
-O.36538
-0.34615
-O.32692
-0.30769
-0.28846
-0.26923
-0.25
-0.23077
-0.21 154
-0.1923 1
-0.17308
-0. 1 53 85
-0. 13462
-0.1 1538
-0.09615
-0.07692
-0.05769
-0.03846
-0.01923
0
0.01923 1
0.038462
0.057692
0.076923
0.096154
0.1 15385
0.134615
0.153846
0.173077
0.192308
0.21 1538
0.230769
0.25
0.269231
0.288462
0.307692
0.326923
0.346154
0.365385
0.384615
0.403846
0.423077
0.442308
0.461538
0.480769
0.5
0.519231

165

81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
2

31.03793
31.105 16
31.03793
30.83621
30.90347
30.83621
30.90347
30.83621
30.83621
30.76898
30.50006
30.231 14
30.3656
30.16391
30.3656
30.02945
29.82776
29.62604
29.49158
29.22266
29.02045
28.74432
28.53723
28.19208
28.4682 1

2142.477 1

——~———__~——~

—

I

—

---

0.538462
0.557692
0.576923
0.596154
0.615385
0.634615
0.653846
0.673077
0.692308
0.71 1538
0.730769
0.75

0.76923 1

0.788462

0.807692

0.826923
0.846154
0.865385
0.884615
0.903 846
0.923077
0.942308
0.961538
0.980769

166

167

Appendixﬂ

Calibration of IPPLUS for Infrared Temperature Measurements
(White, 1996)

This appendix discusses the steps involved in obtaining a calibration between the
IPPLUS image processing system and the Inframetrics thermal imaging system.

EXEERIMENIALEQUIEMENI

A thin (1/8”) Aluminum disc 2 inches in diameter
Minco disc heater 2” inches in diameter
Omega-High Thermal Conductivity Paste
Radiometer and Control Unit

Video Recorder

DC Power Supply

IPPLUS image processing system

Np’SnPP’N.‘

EXEERIMENIALRREBARAIIQN
1. Aluminum disc is painted with black paint to improve emissive properties.
2. Minco heater is attached to disc using paste.

 

 

 

 

 

 

 

 

 

W
C 1 IR “"11”": gm
81.1110 Scanner DISC eater
..__ V v:
'_ fffff - e I
V16 6 ca: sétte DC
Recor er E335;

Figure D1. Experimental Setup

. I

168

W

1. In calibrating the system the radiometer was operated in the point mode. As shown in
Figure B2, the crosshairs were placed near the center of the disk and the cursor
temperature in the text line was used in the calibration.

 

Figure D2. Camera operated in the point mode. The temperature at the location of
the “cross hairs” is indicated in the text line along the bottom of the image.

2. In the ﬁrst calibration test the temperature span was set to ﬁve degrees and the range
was adjusted so the entire ﬁeld of view was dark. Therefore, initially the temperature
of the disk was below the range of the radiometer.

3. Then as the voltage supplied to the heater was slowly increased the output of the
radiometer was stored on video cassette. The test continued until the disk
temperature exceeded the radiometer’s temperature range.

4. The recorded data was then analyzed using the IPPLUS image processing system. As
the temperature of the disk slowly increased, thermal ﬁelds as shown in Figure D2
were randomly captured.

5. Using the line proﬁle or bit map analysis function of IPPLUS the gray scale
intensities of four regions, indicated in Figure D2 with red rectangles, were average
and compared with the cursor temperature. If this is done several times throughout
the temperature range, a plot of temperature versus gray scale can be developed.
Since the output of the radiometer is a standard 8 bit video format, the gray scale
intensities will range between 0 and 255. Therefore, regardless of the temperature
setting the output of the radiometer is divided into 256 discrete levels. This being the

169

case, the slope of a gray scale to temperature calibration curve with radiometer in the
20 temperature range is different than the slope of the of a gray scale to temperature
calibration curve with the radiometer temperature range at 5 degrees. These
differences in slope can be eliminated if the gray scale is plotted as a function of non-
dimensional temperature.

6. A non-dimensional temperature was determined based on the minimum and
maximum
temperature for the particular temperature span setting. For example, in Figure D2

the minimum temperature TL=27.O°C
the maximum temperature TH=127°C .
and the temperature at the location of the cursor is T = 98°C.

T— T
0=——‘—=.71 (B.1)
TH _ TI

7. This non-dimensional temperature is then plotted as a function of the gray scale
intensity and a linear regression can be performed on the data to ﬁnd a relationship
between gray scale and temperature. This plot is illustrated in Figure B3.

8. Steps 1 through 7 are repeated with the temperature span set at 10°C, 20°C, 50°C, and
100°C.

Calibration of IPPLUS (DOS Version)

Non-Dimensional Temperature as a
Function of Gray Scale Intensity .

/

 

 

0 Temperature Range 8 5°C
, ---~- Linear Regression 5°C

/ 0 Temperature Range = 10°C
—— Linear Regression 10°C

0 Temperature Range = 50°C

/ -— Linear Regression 50°C
‘ 0 Temperature Range = 100°C

/ -—- Linear Regression 100°C

9=T-TL/TH-TL
\

 

 

 

I
o 50 100 150 . 200 250
Gray Scale Intensity (0-255)

 

170

Figure D3. Non-dimensional temperature as function of gray scale intensity.

W

Calibrations such as those in Figure D3 illustrate that the gray scale output of the
radiometer is linearly related to temperature. It is interesting that the calibration curve
does not cross through the origin and causes an offset in the relationship between gray
scale and temperature. This offset is a characteristic of the radiometer’s output.

Normally, a black image will have a gray scale intensity of 0, while a white image
will have a gray scale value of 255. Although the output of the radiometer varies
between “black or cold” and “white or hot”, these images. do not produce typical gray
scale values. For example, pixels which are below the temperature range of the
radiometer appear “black”. Normally, the gray intensity of these pixels would be zero.
However, the “black” pixels produced by the radiometer are digitized to values higher
than zero. Therefore, since the “black” values produced by the radiometer are not truly
black an offset exist.

The linear scale used corresponds to : y-intercept = -0.011879
slope = 3.96652 x 10’3

171

 

Table E.l Measured distance from the boron source

The distances were measured from the middle of the boron source to the middle of the
sample.

 

Table E.2 Measured data used for NLIN

I72

 

 

 

 

 

 

 

 

 

 

 

Voltage (volts) Calculated Error in Sample Estimated
Temperature Thermal Number Thermal
Difference (0) Conductivity Conductivity
(5)

1.275 2.45 60.61 2 960
1.375 2.86 56.927
1.048 4.33 34.657 7 653
1.213 5.8 33.174
1.359 7.28 32.494
1.124 3.82 30.764 8 561
1.263 4.822 29.435

1.37 5.67 28.808
1.016 1.79 54.713 6 740
1.203 2.51 46.356
1.203 4.63 55.633 1 975
0.995 3.167 60.708

2.3 1.198 60.092 3 914
1.396 13.9 12.835 17 205
1.285 11.4 12.865
1.021 7.45 13.156
1.151 9.65 10.117 19 263
1.312 12.55 10.014
1.214 10.75 10.059
1.254 2.089 65.088 4 958

 

 

 

 

 

Table E.3 Complete uncertainty analysis

 

 

173

 

‘eroucv‘

'xrgsz‘ngv ,
’ ‘t 11 ~ ‘ .
hmmelluw

 

Figure E.2 Photograph of the jiffy jack

174

 

 

 

Figure E.4 Diamond Sample (silicon side)

175

 

 

Figure E.5 Photograph of Diamond Sample (Diamond Side)

176

 

Figure ESEM results showing a thickness of 2.06 pm

177

 

Figure E. ESEM results showing thickness of 1.70 pm

178

In order to understand if the measured value of emissivity is consistant with
theory for incident radiation striking a surface, transmissivity of diamond was
investigated using a technique described in inframetrics Operator manual. For incident
radiation striking a surface, I + 8 + p = 1, where I: = transmissivity, E = emissivity. and p
= reﬂectivity. An experiment was conducted to measure the transmissivity of diamond

by considering two heat source. The experimental setup consist of:

0 2 Power supplies

0 4 Electrical lead wires

0 The data acquisition system
0 The heat source specimen

A heat source specimen was constructed using four electric wires, each approximately 4
inches long. The insulating material at the end was removed to about 0.5 inches on each
end of the wires. The outer pair of wires are approximately 2.5 cm apart. and the two
inner wires are about 1 cm apart. The two ﬁne wires are about 2.5 mm apart Two ﬁne
wires (approximately 500 microns in diameter), with known resistivity, are connected to

each pair of wires at one end of the stripped electrical wires as shown in Figure 13.] below

 

179

Fine Wine 9411'! lmwn Racism/1m

 

 

 

 

‘1 1'

\

Stripped Electrical Vireo

 

 

Figure E.l Fine wire connection to the electrical wires

The pair of resistors created are bonded to a 0.5 inches by 2.0 inches using Duro super

glue. The heat source specimen is Shown in Figure E.2

Fmﬁs

.1. .L

 

 

 

 

 

 

 

 

 

 

TT I

Figure E.2 Heat source specimen

The electrical lead wires are then connected to the stripped wires in order to create an

electric circuit with each of the ﬁne wires. The setup used is shown in Figure 13.3

180

Fun§§gfikF

 

 

 

 

 

 

 

 

 

 

 

 

Power

Power 51431311.!

Supple

 

Figure E3 The experimental setup

The heat source Specimen is then placed in front of the scanner and known voltages are
applied ( PSl=11, PSZ=6.5) to each of the wires, resulting in a temperature difference
between the two wires. The radiometer settings is then adjusted to the correct inputs

using the control panel. The radiometer settings are:

Temperature Span = 10 OC

Units = LVL

Emittance = 1.0

Center Temperature = Auto
The experiment is then conducted by recording the LVL output for each ﬁne wire using
the point mode function on the radiometer. The level output from each ﬁne wire with the
diamond sample present and without the diamond sample are recorded. The

transmissivity is then calculated from

181

‘L' = ( LVL A - LVL B)with diamond / (LVL A ' LVL B)without diamond

Data from three experimental runs are shown in Table E.1 below

 

 

 

 

 

 

 

 

LVL LVL LVL LVL Transmissivity
without Without with With
Diamond Diamond Diamond Diamond
Test Wire #1 Wire #2 Wire #1 Wire #2
Number
1 4100 5780 3543 4070 0.314
2 4055 5830 3533 3976 0.250
3 4148 5720 3543 4020 0.303
Average 4101 5777 3 540 4022 0.29
Standard 38.0 45.0 4.71 38.4 0.03
Deviation

 

 

 

 

 

 

Table E.1 Results of transmissivity test

The average value of emissivity is consistant with the measured result of .62.

Since reﬂectivity, p, is 1-r+s, it is safe to assume that the reﬂectivity of diamond is about

0.09.

 

 

182

 

—...

 

4W4
7" uuoazeien-aieee

 

 

 

 

f

in i- 1; 11. ii 1'. u a as
We (an-i) -

Figure [5.4 Measurement of transmissivity using FT IR

 

 

 

Figure E4 shows the transmissivity measurement from an FTIR. The result from the
FTIR indicates that the measured average value of 0.29 is in the range given by the

technique used.

BIBLIOGRAPHY

1) Baba, K., Aikawa, Y., and Shohata, N. “Thermal Conductivity of Diamond Film”. 1,
Applihys 69, 7313-7315. 1991

2) Beck, James V., et al. W. Washington. DC:
Hemisphere Publishing Corporation, 1992

3) Beck JV, and Amold. KJ. W John
Wiley and Sons, 1977

4) Beckwith, Thomas G., and Marangoni, Roy D. MgahanioaiMaasnramams 4th Ed.
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5) Berman, R. Ihermalﬂondnotionanﬁolida. Oxford: Clarendon Press, 1976

6) Caber, Paul J ., Martinek, Stephen J, and Niemann, Robert J. “A New Interferometric
Proﬁler for Smooth and Rough Surfaces.” Wyko Corporation. Tucson, AZ. 1996

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8) Graebner, J E. et a1. “Thermal Conductivity in Chemical Vapor Deposited Diamond.“
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9) Graebner J E., et al. “Phonon Scattering 1n Chemical- Vapor-Deposited Diamond.”

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12) Holman, P.J. HeaLIransfer. 2nd ed New York: McGraw Hill Publishing Company.
1990

 

I84

13) Inframetrics 600 Operator Manual, Document # 05250-200 Rev C. Billerico. MA.
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14) Khatami, Saeid. “Microwave Cavity Plasma Reactor Field Map Development and
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15) Kittel, Charles. W 7‘h ed. New York: John Wiley &
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16) Maissel, L I, and F rancombe, M H. Wins. New York:
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17) Malta, D M. et al. “Comparison of the electrical properties of Simultaneously
deposited homoepitaxial and polycrystalline diamond ﬁlms.” LARRLBIIXL 77 (4).
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18) May, Paul W. “ CVD Diamond: A New Technology for the Future.” Endeayom.
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19) Morelli, D T., Beetz, C D., and Perry, T A. “Thermal Conductivity of Synthetic
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20) Nassau, Kurt. “Synthesis of Bulk Diamond. History and Present Status.” Diamond
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21) Ono. A.. et al. “Thermal Conductivity of Diamond Films Synthesized by Microwave

Plasma CVD.” JapanasaLAooLBhysq 25, L808-L810. 1986

22) Pehrsson, Pehr, et al. “Chemical Mechanisms of Diamond CVD.” DiamondEiinuamd
Coatings. Ed Robert Davis. Park Ridge, NJ: Noyes Publications. 1993

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420-429. 1982

24) Reinhard, D. K. Wm. Boston: Houghton

Mifﬂin Co. 1987

25) Van Zant, Peter. W
Emoosaing 2nd Ed. New York: McGraw Hill Publishing Company. 1990

26) Werner, M. et al. “The Relationship between Resistivity and Boron Doping
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185

27) Whallon, Joanne. “The laser Scanning Microscope Laboratory.” Michigan State
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28) White, Matthew. “Development of non-Contact and Non-Destructive Experimental
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Butterworth-Heinemann. 1991

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