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This is to certify that the

thesis entitled

EXPERIMENTAL DESIGN AND TECHNIQUES FOR
MEASUREMENT OF CVD DIAMOND FILM THERMAL
CONDUCTIVITY USING INFRARED THERMOGRAPHY

presented by

SCOTT ALLEN HERR

has been accepted towards fulfillment
of the requirements for

MRS—degree in MECHANICAL ENGINEERING

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ammo-n1

EXPERIMENTAL DESIGN AND TECHNIQUES FOR
MEASUREMENT OF CVD DIAMOND FILM THERMAL
CONDUCTIVITY USING INFRARED THERMOGRAPHY

By

Scott Allen Herr

A Thesis

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

MASTER OF SCIENCE

Department of Mechanical Engineering

1993

ABSTRACT

EXPERIMENTAL DESIGN AND TECHNIQUES FOR
MEASUREMENT OF CVD DIAMOND FILM THERMAL
CONDUCTIVITY USING INFRARED THERMOGRAPHY

By

Scott Allen Herr

Experimental design and techniques were developed for the measurement of the
thermal conductivity of doped CVD diamond films. Three solutions were derived, one
each for one dimensional, two dimensional and radial heat flow. A sensitivity analysis
revealed that the radial heat flow model was most sensitive to the measurement of the
thermal conductivity. Sample parameters such as characteristic length, thickness of doped
diamond and resistivity were chosen from the model to reduce convective effects, obtain
the desired temperature rise and minimize the uncertainty in the estimation of the thermal
conductivity.

Two diamond film samples were made according to the specifications determined
by the analytical analysis. Both samples consisted of doped and nondoped layers on the
topside and were chemically etched from the backside, leaving a free standing diamond
diaphragm 3 mm in diameter. The thickness of the doped and nondoped films were
approximately 5.6 pm and 1.0 pm, respectively, for'both samples.

A new experimental setup was designed and constructed. An infrared imaging
temperature acquisition system was implemented to improve on the spatial, temporal and

mechanical limitations of contact sensors such as thermocouples and resistance thermometers.

Preliminary results for the thermal conductivity of a semiconducting diamond film
were obtained from five experiments using the method of least squares to minimize the
error between the measured temperatures recorded by the infrared temperature acquisition
system and the calculated temperatures determined by the optimal radial heat flow model.
The thermal conductivity along with the experimental uncertainty was determined to be
249 i 13 W/m K.

These are the fast reported values for a semiconducting diamond film. The mean
value falls within the range reported for undoped diamond films (190-1350 W/m K). The
experimental uncertainty for this method (approximately 5%) is also the first to be

determined utilizing uncertainties in all measured experimental parameters.

to my parents Jack and Jan

iv

Acknowledgments

I would like to thank my co-advisors, Professor J. V. Beck and Professor J. J.
McGrath for their guidance in the present work. I am greatful for the opportunity to learn
from their example as both researchers and engineers.

I would like to personally thank Professor J. V. Beck for the opportunities he has
extended to me as his research assistant in the Thermal Properties Measurement
Laboratory over the past two years. This opportunity has presented several rewarding
experiences; one of which was working with Dr. Bertrand Gamier from Laboratoire de
Thermocinetique-ISITEM, Nantes, France. Dr. Gamier’s creative thinking, meticulous
explanations and exceptional experimental talents have been an important influence in my
own laboratory techniques.

I would also like to thank Professor C. W. Somerton for his review of the present
work. I believe Professor Somerton to be an exceptional teacher and consider the time
spent in his classroom to be an accelerated learning experience.

Many thanks go out to Professor M. Aslam and S. Sahli for their expertise in
diamond deposition and microstructural fabrication; their help was crucial in designing
the optimal diamond semiconductor for our experiments. I wish to also thank C.
Mindock for her involvement in the chemical etching process.

A special thanks goes out to my undergraduate advisor and friend Professor H. M.

Dixon who’s inspiration throughout the years has elevated my understanding of physics

and engineering. Professor Dixon is a most remarkable scientist and educator and I am
greatful for his continual interest and involvement in my education.
Last but not least, I want to thank my family for the support and encouragement

they have given me throughout my life.

Table of Contents

List of Tables
List of Figures

Nomenclature

Chapter
1 Introduction
1.1 Objectives
2 Background and Literature Review

2.1 CVD Diamond Films
2.2 General Experimental Methods Determining Thermal Properties

2.3 Special Experimental Methods for CVD Diamond Films

3 Problem Description
3.1 Mathematical Modeling
3.1.1 One Dimensional Heat Conduction Model
3.1.2 Two Dimensional Heat Conduction Model

3.1.3 Radial Heat Conduction Model

vii

xi

XV

21

21

25

28

32

Design of the Diamond Film Sample

4.1 Determining the Best Experimental Model

4.2 Special Requirements for the Desired Temperature Rise
4.2.1 Volumetric Heat Generation in Diamond
4.2.2 D0ping Requirements for the Diamond Film

4.3 Limiting Natural Convection

4.4 Analytical Results

4.5 Topside Sample Fabrication

4.6 Bottomside Sample Fabrication

Experimental Techniques

5.1 The Data Acquisition System
5.1.1 The Inframetrics Model 6OOL IR Imaging Radiometer
5.1.2 Thermal Image Processing System
5.1.3 The Omega 880 Digital Multimeter

5.2 System Calibration Check

5.3 Experimental Setup

5.3.1 Preparation of the Sample Setup

Procedures and Results for One Dimensional Radial Experiments
6.1 Determining the Emissivity
6.2 Measuring the Electrical Power

6.3 Radiometer Settings

viii

35

36

50

50

52

56

61

63

69

73

73

74

77

78

79

84

84

86

86

89

91

6.4 Surface Temperature Measurements
6.4.1 Capturing Images Using the Thermal Image Processor
6.4.2 Temperature Acquisition
6.5 Measurement of the Thermal Conductivity
6.5.1 The Simplified Experimental Model
6.5.2 Estimation of the Thermal Conductivity

6.6 Experimental Uncertainty

7 Summary and Conclusions

Suggestions for future work

List of References

Appendices
Appendix A Calibration of the Inframetrics 600L Radiometer for Rapid
Transient Thermal Events
Appendix B Temperature Distribution for Experiments 1-5
Appendix C Program NLIN

Appendix D NLIN Output File for Experiment #5

93

93

98

98

99

100

111

115

118

120

123

123

132

137

143

Table 2.1:

Table 2.2:

Table 4.1:

Table 6.1:

Table 6.2:

Table 6.3:

List of Tables

Thermal conductivity of diamond film evaluated at different
thicknesses at room temperature (Morelli et al., 1988).

Thermal conductivity parallel to the film surface as a function
of film thickness (Graebner et al., 1992).

Optimum sample design parameters assuming a thermal
conductivity of 1000 W/m K.

Temperature vs. resistance for sample A.

The thermal conductivity estimated by NLIN for each of the
five experiments at different power levels.

Experimental parameters. their uncertainties and the corresponding
contributions to the uncertainty of the thermal conductivity.

11

14

62

90

110

112

Figure 2.1:
Figure 2.2:

Figure 2.3:

Figure 2.4:

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 3.1:

Figure 3.2:

Figure 3.3:

Figure 3.4:
Figure 3.5:

Figure 3.6:

Figure 4.1:

Figure 4.2:

Figure 4.3:

List of Figures

The diamond cubic unit cell.
Schematic of a deposited diamond film sample.

Microscopic view of CVD diamond film prepared by the
hot filament method.

Microscopic view of CVD diamond film prepared by the
microwave method.

Thermal conductivity parallel and perpendicular to the
film surface.

A block diagram of the type of experimental setup used by Albin,
Winfree and Crews ( 1990) to determine thermal diffusivity.

Schematic of the experimental setup for the photothennal laser beam
deflection technique (Machlab et al., 1991).

Ratio of the diamond film thickness to the silicon substrate thickness
in the z direction.

Schematic of the one dimensional model.

Schematic of the two dimensional diamond film sample etched from the
backside.

Schematic of the two dimensional model.
One of the symmetric quadrants for the two dimensional model.
Schematic of the radial heat conduction model.

The non-dimensional sensitivity coefficient as a function of characteristic
length for different film thicknesses, 1D case.

The non-dimensional sensitivity coefficient as a function of characteristic
length for different temperature rises, 1D case.

The non-dimensional sensitivity coefficient as a function of characteristic
length for different film thicknesses, 2D case.

10

18

19

24

26

29

30

31

33

39

40

42

Figure 4.4:

Figure 4.5:

Figure 4.6:

Figure 4.7:

Figure 4.8:

Figure 4.9:

Figure 4.10:

Figure 4.11:

Figure 4.12:

Figure 4.13:

Figure 4.14:

Figure 4.15:

Figure 4.16:

Figure 4.17:
Figure 4.18:
Figure 4.19:
Figure 4.20:
Figure 4.21:

Figure 4.22:

Comparison of the non-dimensional sensitivity coefficient between the one
dimensional and two dimensional cases at the same characteristic length

The non-dimensional sensitivity coefficient as a function of characteristic
length for different temperature rises, 2D case.

The non-dimensional sensitivity coefficient as a function of characteristic
length for sample widths of 1.0, 0.50 and 0.25 cm, 2D case.

The non-dimensional sensitivity coefficient as a function of diameter for
different film thicknesses, radial case.

The non-dimensional sensitivity coefficient as a function of diameter for
different temperature rises, radial case.

Comparison of the non-dimensional sensitivity coefficient as a function of
characteristic length between the 1D, 2D and radial cases.

Schematic of a sample with 3 doped diamond film, a layer of SiO2 and a
silicon substrate.

Schematic of a samme with doped and undoped diamond films, a layer of
SiO2 and a silicon substrate.

Schematic of the electrical circuit.

Temperature rise as a function of applied voltage for different film
thicknesses.

The non-dimensional sensitivity coefficient as a function of
(tnf * characteristic length) for the 1D, 2D and radial cases.

Ratio of natural convection to the total heat generated in the doped film as a
function of diameter for different film thicknesses.

The non-dimensional sensitivity coefficient as a function of diameter for
different doped and undoped film thicknesses.

Photograph of the hot filament CVD diamond deposition chamber.
Thickness of the undoped film on sample A measured in A by the SEM.
Thickness of the undoped film on sample B measured in A by the SEM.
Thickness of the doped film on sample A measured in A by the SEM.
Thickness of the doped film on sample B measured in A by the SEM.

The front side of sample A.

xii

43

45

47

48

49

51

51

53

55

57

59

65

65

66

68

69

70

Figure 4.23:

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 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:

The diamond mask pattemed on the back side of sample A.
Schematic of the data acquisition system.
The 600L Infiared Radiometer.

The 6" close-up lens and the 3x telescope lens used to increase the spatial
resolution of the radiometer.

The Omega 880 Digital Multimeter.

Corrected temperature of the thermocouple as a function of measured
temperature of the radiometer.

Heater surface temperature as a function of surface location.
Schematic view of the sample afier experimental preparation.
Photograph of the sample after experimental preparation.

Setup used to determine the emissivity of the diamond film for sample A.

The relationship between the temperature measured by the radiometer and the
temperature measured by the thermocouple at an emissivity setting of 0.63.

Sample setup viewed from the front side.
Sample setup viewed from the back side.
Image 1 from experiment #1 at 0.40 watts.
Image 2 from experiment #2 at 0.62 watts.
Image 3 from experiment #3 at 0.88 watts.
Image 4 from experiment #4 at 0.40 watts.

Image 5 from experiment #5 at 0.88 watts.

Measured and calculated temperature distribution for experiments 1, 2 and 3.

Measured and calculated temperature distribution for experiments 4 and 5.
Residual distribution for experiment #1, 107 pts.
Residual distribution for experiment #2, 107 pts.

Residual distribution for experiment #3, 107 pts.

xiii

71

74

75

77

79

80

83

85

85

87

88

94

95

97

97

97

97

97

102

102

103

104

104

Figure 6.15:
Figure 6.16:
Figure 6.17:
Figure 6.18:
Figure 6.19:
Figure 6.20:

Figure 6.21:

Figure A.1:
Figure A.2:

Figure A.3:

Figure A.4:

Residual distribution for experiment #4, 121 pts.

Residual distribution for experiment #5, 121 pts.

Sequential estimation of the thennal conductivity for experiment #1, 107 pts.
Sequential estimation of the thermal conductivity for experiment #2, 107 pts.
Sequential estimation of the thermal conductivity for experiment #3, 107 pts.
Sequential estimation of the thermal conductivity for experiment #4, 121 pts.

Sequential estimation of the thermal conductivity for experiment #5, 121 pts.

Setup used to calibrate the infrared radiometer to record transient events.
Vertical refresh signals resembling legitimate line scans.

Thermocouple termperature as a frmction of radiometer voltage for the six
different calibration runs.

Radiometer voltage recorded in fast line scan mode during calibration.

xiv

105

105

107

107

108

108

109

124

129

130

131

”@300

Nomenclature

Thermal Conductivity
Emissivity

Thermal Diffusivity
Specific Heat

Density

Temperature

Thickness of Doped Film
Thickness of Undoped Film
Convection Coefficient
Temperature Difference
Relative Temperature Rise
Area

Volume

Least Squares Function
Sensitivity Coefficient Matrix
Voltage

Electrical Resistance
Thermal Resistance

Time

Volumetric Heat Generation
Convective Fin Term
Eigenvalue

Cylindrical Coordinate

XV

DU‘IQ'D“[‘-<O‘

Radius

Nondirnensional Sensitivity Coefficient
Length

Characteristic Length

Resistivity

Stefan-Boltzmann Constant

Errrissive Power

Heat Flux

Uncertainty in the RelativeTemperature Rise
Uncertainty in the Thermal Conductivity
Measured Temperature

Calculated Temperature

Width

Standard Deviation of Thermal Conductivity
Standard Deviation of Temperature

t statistic

Prandtl Number

Rayleigh Number

Hertz

Angstroms

Micrometer

Analog to Digital Conversion

xvi

[ml

[m]

[m]
[Q-cm]
[W/m2 K‘]
[W/m2]
[W/mzl
[°C]
[W/m K]
[°Cl

[°C]

[In]

[W /m K]
1°C]

[cycles/sec]
[10‘10 m]
[10“ m]

Subscripts / Superscripts

eff
rad
x,y,z
m,n,j
corr
meas
hcap
b

cond

Effective
Radiation
Directions
Surroundings
Variable Indices
Corrected
Measured

Heat Capacity
Boundary
Conduction
Cross-Sectional
Actual

Parallel
Perpendicular

xvii

Chapter 1

Introduction

For centuries the diamond has captivated both spectator and scientist alike. From
its brilliant sparkle and unequalled hardness, the diamond has long been a symbol of
strength and perfection. Though desirable and intriguing in its physical makeup, nature
alone held the secret to its existence, and its rarity limited its use. However, with recent
discoveries in chemical vapor deposition (CVD), diamond has been transformed from its
rare jewel status into a promising engineering material.

Diamonds are used in a wide variety of applications such as cutting and grinding
tools, bearings and gears, biological protheses, speakers, amplifiers, telecommunications
systems, lasers, computer hard disks, integrated circuits, and transducers. Since diamond
has the highest known thermal conductivity of 2600 W/m-K, six times that of copper and
eight times that of gold, it also makes an ideal heat sink. For example, diamond would
provide a means of dissipating large amounts of potentially damaging heat generated
when integrated circuit chips are mounted in a dense configuration. Higher density
integrated circuit configurations could form the basis for computers operating four orders
of magnitude faster than the present-day types (Spehar, 1991).

Through its unique serrriconduction capabilities and its "second-to-none" structural

1

2

integrity, significant contributions could also potentially be found in thermal sensors.
Since diamond will resist corrosion, oxidation, diffusion, adhesion, chemical attacks,
intense pressures and extreme temperatures, it makes a suitable sensor for temperature
measurements in the harshest environments. In fact, only hot carbide-forming metals or
strong oxidizing agents such as molten sodium nitrate will affect it. Applications might
also be in hypersonic flight, nuclear reactors, and high temperature industrial processes.

In order for diamond to become a semiconductor, however, it must be doped with
a material such as boron during synthesis. Boron allows the diamond crystals to become
charge carriers and the doping levels can be varied depending on the desired resistivity.
Since the added impurities can affect the behavior of diamond, the thermal properties of
doped diamond films must be determined to insure the proper performance for the
particular application.

However, doped diamond films pose several problems to traditional experimental
methods used to determine thermal conductivity. Its micro-structural size and rapid
thermal response limit both the thermocouple and the resistance thermometer as reliable
temperature measurement devices. As a result, we turn to an infrared thermography
temperature acquisition system which can image microstructures and track rapid thermal
responses to obtain the necessary surface temperature data to estimate the thermal

conductivity.

1.1 Objectives:

The general objective of this research was the development of analytical tools and
experimental methods which would allow thermal analysis of micro-structures. The
primary objective of this work was to utilize these methods to determine the thermal
conductivity of doped diamond films, which were manufactured by the Micro-Structures

Laboratory at Michigan State University. In particular the objectives were:

1) to determine which experimental model yields the optimal estimation of the
thermal conductivity,

2) to utilize analytical tools to determine the optimal design of the diamond
specimens,

3) to prepare at least one "masked" diamond film specimen which corresponded to
the optimal design specifications,

4) to design a suitable experimental set-up for accurate temperature measurements,

5) to implement the infrared thermography and high resolution image processing
system to image the surface temperature of the film and determine the relative
temperature rise,

6) to extract temperature data from the resulting thermal images and estimate the
thermal conductivity (k) of a doped CVD diamond film sample.

Chapter 2

Background and Literature Review

Perhaps the area of greatest potential for diamond films is in commercial and
military electronics. With its remarkably high thermal conductivity, diamond films have
already become an attractive material for applications as heat sinks and heat Spreaders for
various electronic components. In these applications, diamond films are used to alleviate
the damaging effects of self-heating. They can deliver manufacturers of
telecommunications, computers, and integrated circuit chips a much improved means of
dissipating heat, boosting both component reliability and performance. A diode laser, for
example, producing a beam which gives a line heat source a few microns wide can be
cooled effectively by mounting it against a diamond heat spreader (Graebner et a1. 1992).
In addition to being an "ideal" heat spreader for high energy density lasers, diamond has
the potential itself to become a laser. A diamond laser core would withstand extreme
high thermal stress due to its ability to efficiently conduct heat, decreasing heat expansion
and ultimately increasing laser efficiency (Spehar 1991). However, defining the limits
of applicability for diamond films has required an understanding of the thermophysical

characteristics of the material.

2.1 CVD Diamond Films

Diamond films, like the diamond gem, consists of carbon atoms bonded together
in strong sigma type convalent bonds. The atomic orbitals are sp3 and the bonds formed
are very strong. The carbon atoms are arranged in a tetrahedral formation resulting in the

diamond cubic unit cell seen in Figure 2.1.

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1. The Diamond Cubic Unit Cell

The structure of diamond gives rise to extreme hardness with the ability to withstand
pressures of over 900,000 lbs/inz. The stiffness of these unique molecular bonds also
contribute to diamond’s high thermal conductivity. In nonmetallic materials, heat
conduction occurs primarily through a phonon to phonon transfer phenomenon. This
mechanism, which is dependent on the molecular vibrational energy within a material, is

most efficient when atomic bonds are stiff and is dampened when bonds are elastic.

6
Through a process known as chemical-vapor deposition (CVD), polycrystalline

diamond films can be deposited on non-diamond substrates, such as silicon, from carbon

enriched gas mixtures, see Figure 2.2. The two most commonly used CVD methods for

 

diamond film

 

silicon substrate

 

 

 

 

 

Figure 2.2. Schematic of deposited diamond film sample.

heating the surface of the substrate to induce diamond growth are known as the hot—
filament assisted method, which uses electrically resistive heating, and the plasma
enhanced microwave method. The hot-filament heating provides higher quality diamond
films but at low deposition rates, between 05-10 rim/hr. A photograph of the surface of
a diamond film prepared by the hot-filament method is viewed in Figure 2.3. On the
other hand, the microwave heating method provides a higher deposition rate, usually
between 1-4 rim/hr, but at a lower quality film A photograph of the surface of a

diamond film prepared by the microwave heating method is seen in Figure 2.4.

 

  
  

 

Figure 2.3. Microscopic view of
CVD diamond film prepared by
the hot filament method.

i"

lSKU memos tam

 

Figure 2.4. Microscopic view of
CVD diamond film prepared by
the microwave method.

8

However, even when using the best CVD system one must be aware that a certain
amount of impurities and defects normally are present as a result of this process.
Although research into the CVD process over the past few years has increased the quality
of the diamond, impurities still have a major effect on the physical properties of diamond,
especially the thermal conductivity. For example, Baba et a1. (1991) reported that by
increasing the methane (CH, ) concentration fiom 1% to 5% the thermal conductivity
decreased from 1200 W/mK to 200 W/mK This reduction was assumed to be the result
of phonon scattering due to the hydrogen impurity found in the films. The thermal
conductivity can also be severely affected by the structural defects of the film. Bad grain
boundaries can result in poor thermal contact within the films. Since CVD diamond films
are prone to impurities and small defects, its application as a heat sink could be quite

restrictive without experimentally determining the thermal conductivity.

2.2 Experimental Methods Determining Thermal Properties

Generally, methods of determining the thermophysical characteristics of materials
are divided into three groups: steady state methods, quasi-stationary methods, and
transient methods (Vorobei 1986). The estimation of the thermal conductivity is not
limited to any one of these methods. However, the estimation of thermal diffusivity or
volumetric heat capacity requires transient experiments. The phenomenon of heat
conduction is simply described by heat flowing down a temperature gradient to restore
thermal equilibrium. Thus, it is necessary to introduce a heat source to perturb the
thermal equilibrium in the material of interest to establish a temperature gradient. The

temperature distribution and the applied heat must be measured in order to estimate the

9
thermal properties. The mathematical description will be discussed later in Chapter 3.

There are basically two different types of experimental analysis currently in use,
the absolute method and the relative method. The absolute method provides a very
accurate means of estimating the thermal properties but it requires expensive
instrumentation (e.g. thermocouples, resistance thermometers, IR cameras, heaters, etc.)
and great care in their use. In the relative method the thermal conductivity is determined
by reference to a material of known conductivity. Due to the desired accuracy of the

present analysis, the absolute method is employed in this work.

2.3 Special Experimental Methods for CVD diamond

There have been several important experimental studies concerning the thermal
conductivity of CVD diamond films over the past several years. The different
experimental techniques used in these works can be divided into four different categories:
surface instrumentation, the laser pulse technique, IR radiation thermography and
phototherrnal laser beam deflection. These techniques and experimental procedures
which are instrumental in determining thermal conductivity both parallel (k.,) and
perpendicular (k D to the film surface are discussed in turn in the following paragraphs,

see Figure 2.5.

10

 

 

F dilamond lem

 

 

 

 

Figure 2.5. Thermal conductivity parallel and perpendicular to the film surface.

Surface Instrumentation:

Surface instrumentation (such as, thermocouples, thermistors and resistance
thermometers) is currently the most common form of detecting and recording the
temperature distribution on the surfaces of diamond films. Morelli, Beetz and Perry
(1988) introduced an experimental procedure known as the steady state four-probe
technique to determine the thermal conductivity parallel (k..) to the surface of the film as
a function of thickness and temperature. This method involves the use of a thin film
heater and four thermocouples. The heater which was attached to one end of the diamond
film sample generated a thermal gradient across the film surface. The thermocouples
which were also attached to the surface using conductive silver paint monitored the
temperature distribution. The substrate was etched away from the backside resulting in
a window 10mm x 5mm of free standing diamond. The sample was then placed in a
vacuum so that convection could be neglected. The effects of radiation were minimized

by enclosing the sample between two metal shrouds. The heat input and the temperature

11

distribution were recorded at steady state and the thermal conductivity was determined
as a function of temperature and thickness. The absolute uncertainty in their results
(~15%) was determined by the accuracy to which the film thickness could be measured.
The relative uncertainty was determined from the resolution of the thermocouple voltage

(~1-2%) at room temperature. Table 2.1 displays some of their experimental results.

  

Thickness (pm)
r 300 I
10
100

  
  
   

 

  
 

 

 

 

Table 2.1. Thermal conductivity of diamond film evaluated at
different thicknesses at room temperature, Morelli, Beetz and
Perry (1988). Films were prepared using the hot filament method.

Anthony, Fleischer, Olson and Cahill (1991) estimated the thermal conductivity
both parallel (k,.) and perpendicular (k i ) to the surface of a diamond film. The sample,
which had dimensions 3.3cm x 1.5 cm x 170 pm, was also prepared by the hot-filament
CVD method. In their study, two different techniques which utilized surface
instrumentation were used. Parallel to the surface the thermal conductivity was measured
between 6-100 K using the steady-state four-probe technique. A 1000-W CuNi thin film
heater was attached to one end of the sample to induce the temperature gradient and
carbon resistance thermometers, which were attached to the surface of the sample,

recorded the temperature distribution. In the perpendicular plane, k i was determined

12

between 80-300 K using what they termed as an "ac diffusive heat wave technique".
Here a silver line 100 pm wide and 3000 A thick was deposited directly on a silicon
substrate to act as a heater-thermometer. By knowing the thermal conductivity of the
substrate and by measuring the temperature at the opposite face they calculated k L. From
their study Anthony and Fleischer concluded that the thermal conductivity of diamond
was a strong function of temperature. They also found that k L > k u but were unable to
positively conclude this due to experimental enors associated with each method.

Graebner, Jin, Kammlott, Herb and Gardinier (1992) used a similar setup to
determine the thermal conductivity parallel to the film surface (k u) using two thin film
heaters which were evaporated directly onto the sample, one near each end. The
temperature distribution was measured again by a row of four very fine thermocouples.
Measurements were performed on several samples prepared by the microwave CVD
method. The samples having different thicknesses were placed in a vacuum where the
effects of convection and radiation were neglected. From their study Graebner, Jin and
Kammlott also concluded that the thermal conductivity was a function of the film
thickness.

Graebner, Mucha, Seibles and Kammlott (1992) determined the thermal
conductivity using a technique that involved etching a window of free standing diamond
2x4 mmz, similar to the Morelli, Beetz and Perry (1988) procedure. The remaining
silicon served as a rugged platform to support the film as well as a heat sink and a
referenced temperature boundary. The flow of heat from a heater in the center of the
window was monitored with thermocouples. The thin film heater and the chromel and

constantan thermocouples were deposited on the surface of the diamond by standard

l3

evaporation techniques. The steady state temperature distribution, detected by the
thermocouples, was then compared with a numerical simulation to extract the thermal
conductivity parallel to the surface (k u)- Their efforts yielded values for k n in the range
of 200-600 W/m°C. The choice of window dimensions was made to reduce the effects

of radiation. To predict the thermal influence from radiation they developed the ratio,

km, ~ Zoeszg 2 1

kcond K t

where km is the conductance of heat along the film of thickness t, km is the effective
conductance due to radiation between the film and its surroundings, 0' is the Stefan-
Boltzman constant and K is the thermal conductivity. By evaluating this ratio under
unfavorable conditions 8=1, To=300 K, t = 2pm and x = 300 W/m K and varying the
width (w), they were able to neglect the effects of radiation when the ratio became
significantly small. The effects of convection were not considered in their study. The

results of their study are presented in the Table 2.2.

14

  

Thickness (pm)
2.8

   

 

  
 

 

  
 

3.8

 

7.0

l 7.3

7.5

  
  

 

 

7.8

 

7.8

 

10.6

 

11.6

 

13.1

 

 

 

Table 2.2. Thermal conductivity parallel to the film surface (k [I)
as a function of film thickness presented by Graebner, Mucha,
Seibles and Kammlott (1992)

Baba et al. (1991) concluded that the thermal conductivity for diamond films was
also dependent on the amount of hydrogen impurity deposited from the methane (CPL)
gas mixture. In this study an experimental method termed the "ac calorimetric" method
was employed. Here, one end of the diamond samples was periodically heated by a
halogen lamp. The temperature amplitude (TAG) at a distance (x) away from the heat
source was monitored by a thermocouple which was attached to the sample’s surface. A

relation between TAC and distance (x) is given by

15

= _q__ - 32‘
lanacl ln<4rtfdcp) ( a )x 2.2

where q is heat quantity, f is the heating frequency, and C,, d and or are the specific heat,
thickness and the thermal diffusivity of the film. The thermal conductivity was extracted
from the diffusivity of two samples and evaluated from the slope of the above equation
by varying x. The thermal conductivity of the film prepared with 1% CH4 concentration
reached 1200 W/m K but the thermal conductivity of the film prepared with 5% CH4
concentration decreased to less than 200 W/m K. Both samples were hot-filament CVD

films.

Laser Pulse Technique:

While it is relatively straightforward to measure It" using surface instrumentation,
it is much more difficult to measure k L perpendicular to the film surface due to the small
thermal resistance in this direction. For this reason a non-contact method which uses
laser pulses to heat the face of the film and fast thermometry to monitor the arrival of the
thermal wave at the opposite face is employed. The thermal conductivity can then be
related to the short duration thermal shock applied by the laser.

Graebner, Jin, Kammlott, Bacon and Seibles (1992) utilized this laser pulse
technique to help determine whether any anisotropic behavior existed in their CVD
diamond films. In their study the thermal conductivity parallel to the surface kn was
determined using standard surface instrumentation procedures previously discussed and
the thermal conductivity perpendicular to the surface k l was determined by using a Q-

switched Nd:YAG laser as a heat source. The sample was glued with silver paste over

16

a hole to a temperature controlled copper plate. Germanium lenses were used to collect
thermal radiation and measure the temperature on the back side of the sample. The heat
from each laser pulse conducted laterally through the sample to thermal ground at its
edges. The temperature rise on the back surface and the rapid transient after each pulse
were recorded by the system. The thermal jump AT(t) was calculated fiom the analytical

expression presented by Parker et al. (1961)

 

AT(t) =A[1+22 (-l)”exp( '12:"2atn, 2.3
n=1

where A=q/ng and a=kjlpCP; q is the absorbed enery per unit area and a is the
thermal diffusivity. Equation 2.3 is used to solve a system of equations where A and or,
are the unknown quantities. By solving for the thermal diffusivity and measuring both
the thermal response and the characteristic length of the film. the thermal conductivity,
k l, was calculated assuming a value for pCp. A conductivity k L of 800 and 1210 W/m
K was found for two different samples with thicknesses of 234 and 144 pm respectively.
Values for k L were observed to be at least 50% greater than values obtained for kII .
Later the same year Graebner, J in, Kammlott, Herb and Gardinier utilized a similar
setup to determine k L as a function of thickness. Here the film surface was heated using
a periodic laser pulse. The temperature at the top surface was monitored using a high
speed infrared detector. Four samples, 0.5 x 1.0 cm2 in area, with average thicknesses
ranging between 28-408 pm were evaluated. Their results concluded that the average k i

through the film increased from 1000 to 2100 W/m K as the thickness increased.

17
Infrared Thermography:

Ono, Baba, Tunomoto and Nishikawa (1986) utilized this non-contact method to
determine the thermoconductivity parallel to the film surface. Using a long diamond film
sample suspended by heated supports in a vacuum, the temperature distribution along the
length was measured by an infrared thermograph. The surface area of the samples
analyzed was 20 mm x 5 mm while the thicknesses varied between 7-30 pm.
Measurements were made between 100 °C and 130 °C on microwave plasma CVD
diamond. From the results of this study, the thermal conductivity of the diamond films
increases rapidly with decreasing concentration of methane. The highest value for It" was
approximately 1000 W/m K.

A significant study deterrrrining the kll of diamond films using IR thermography
was later done by Albin, Winfiee and Crews (1990). The thermal conductivity was
extracted from the measurement of the thermal diffusivity. Periodic heating was provided
by a 20W, 1.064 pm Nd:YAG laser and the time dependent surface temperature was
measured by a 8-12 pm infrared camera. A diagram of the set-up is displayed in Figure

2.6.

18

 

- \

 

In Gamma

 

 

I
\ I
Focusing Mlnor Sm”

ImageProcessor

 

 

 

Figure 2.6. A block diagram of the type of experimental setup
used by Albin, Winfree and Crews determining thermal
diffusivity.

Temperature measurements between 25-35 °C were made on the back side of the sample
using the infrared camera. The camera scanned a single horizontal line which passed
through the center of the sample and the heating area. An image processor was used to
digitize 128 successive images. Each image was compressed into a single temperature
profile resulting in a sampling rate of 1/30 of a second. The IR camera allowed for a
temperature resolution of less than 0.02 °C and a spatial resolution of better than 1 mm.
In this study k.' for two samples of thickness 16 and 32 pm were determined to be 1350
and 1328 W/m K respectively. The advantage of this technique is that the thermal
diffusivity and the thermal conductivity of diamond films can be determined without

special sample preparation.

19
Photothermal Laser Beam Deflection:

The technique known as photothermal laser beam deflection was introduced by
Machlab, McGahan and Woolham (1991) as an alternate method of determining k. in
diamond films. This technique also known as the "mirage effect" uses the assistance of
two separate laser beams. A diagram of the experimental setup used by Machlab,

McGahan and Woolham is displayed in Figure 2.7.

 

 

 

 

 

Figure 2.7 . Schematic of the experimental setup for the
Phototlrermal Laser Beam Deflection Technique, Machlab
et al., 1991.

One beam generates heat pulses within the sample producing heat pulses in the air above.
The thermal pulse in the air results in an optical index of refraction gradient. The second

beam passes through the index of refraction gradient and is deflected with components

20
both in the plane and perpendicular to the plane of the sample. Since the heating beam

is periodic, the wavelength of these propagating waves can be detected. Because the
wavelength of these thermal waves depends on the frequency of the heating beam and on
the thermal properties, the thermal conductivity kll can be obtained. Although their study
did not include experimental results for diamond films, they were successful in
determining kII for aluminum. This method like the laser pulse technique and IR radiation
thermography offers a non-destructive experimental evaluation of the thermal properties
in CVD diamond films.

Although there have been several studies on the thermal conductivity of diamond
films in recent years, they have focused on non-doped films for heat sink applications.
Presently, very little is known about the effects boron doping will have on the properties
of diamond. As a result, the author was unable to find information concerning the
thermal characteristics of doped diamond films.

Without an understanding of how the doping process affects the thermal properties
of CVD diamond films, applications as a semiconductor could be restricted and
unreliable. As a result, experimental methods and analytical tools are presented in the
present work to aid in the determination of the thermal conductivity of doped diamond

films.

Chapter 3

Problem Description

In the area of inverse heat conduction, which includes the estimation of the
thermal properties of a material from the relative temperature distribution and temperature
rise, it is imperative that an accurate mathematical model of the system be developed.
The mathematical model is used to describe the experimental setup such that the thermal
behavior of the experiment can be predicted analytically. Without such a model, the
thermal properties cannot be determined and without an accurate description fiom the
model, any results are meaningless. Since there is a certain amount of error and
uncertainty associated with different experimental setups, the role of the mathematical
model can also become instrumental in pre-determining the overall success of an
experiment. This section is dedicated to the discussion of three different mathematical
models considered in the present work. The models will later be analyzed in Chapter 4

to determine the "best" experimental setup.

3.1 Mathematical Modeling
The phenomenon of heat conduction is described by the energy of motion

between adjacent molecules. In nonmetallic solids, molecules having greater energy and

21

22

motion translate their energy to adjacent molecules at lower energy levels. In a solid

body with a temperature gradient, Fourier’s Law is used to relate the heat flux (q) to the

g(r, t) =-k(r, t)VT(I, t) 3-1
temperature (T) where the tensor k [W/m K] is the effective thermal conductivity of the
material, the temperature gradient VT [“C/m] is a vector normal to the isothermal surface
and the heat flux vector q [W/m’] is the heat flow per unit time and unit area. The minus
sign is inserted in accordance with the second law of thermodynamics. For example, if
heat flows in a positive direction, the temperature must decrease in that direction. The

three components of q in the x, y and 2 directions are given by

__ 6T 6T __ kaT
qx" xTxlqy= -ky'-_ yay IM-qz ‘kzg- 3’2

for an orthotropic solid (Carslaw and Jaeger, 1959). The accurate estimation of the

thermal conductivity k, involves a solution to the heat conduction equation,

fans. (‘Q’D) dA+fc.v_g(r’ t) dV=fM PC p g—gdv 3.3

which is derived from the conservation of energy (Beck et a1. 1992). The first term
represents the net heat flux rate into the control volume; g(r,t) is the rate of internal
volumetric heat generation; and the right side of the equation represents the energy
storage rate. The solution of the equation in this study is solved for the following general

assumptions:

23
1) heat dissipated through natural convection,

2) insulated at edges

3) radiation neglected

4) no energy storage (steady state condition),
5) isothermal in the z direction; q2 = 0,

6) diamond film is isotropic.
Under these assumptions, general steady state heat conduction reduces to,

fans. ('Q‘D) dA+f g(r' t) dV=0 3.4

GOV.

Steady state experiments were chosen to be modeled instead of transient
experiments due to the performance limitations of the experimental equipment. Although
the data acquisition system, comprised of the infrared camera and the image processor,
had excellent spatial resolution when viewing still thermal images, the image processor
software is not equiped for imaging transient events. In light of this, another data-
acquisition system, comprised of the infrared camera and the National Instruments AT-
MIO-16F5 AID board, was developed by the author. Using this system transient thermal
events can be acquired, however, the spatial resolution is poor since the infrared camera’s
sampling rate is nearly ten times that of the fastest allowable sampling rate on the
available A/D board. A detailed description of both data acquisition systems is given
later in Chapter 5 and Appendix A.

Significant amounts of heat dissipated through convection and radiation can cause

 

24

severe problems in the estimation of the thermal conductivity of microstructures such as
CVD diamond films. For this reason, analytical tools are presented later in Chapter 4 to
help aid in the minimization of these errors and to better define what "significant" means
in regards to the experimental results.

In order to improve our experimental results to determine the thermal conductivity
k“, which is the effective thermal conductivity parallel to the diamond film surface, we
must maximize the heat conduction in the film and minimize the heat conduction in the

silicon substrate parallel to the film surface, see Figure 3.1.

 

 

 

 

 

Figure 3.1. Ratio of the diamond film thickness to the silicon thickness in
the z direction.

By comparing the thermal resistance of the diamond to the silicon in the parallel plane

we see that,

Rd = ksibsi 35
R91 kdbd

 

25
where Rt] is the thermal resistance and 8” is the thickness of each layer. With k,, = 150

W/mK, k“ = 1000 W/m K, d = 400 pm and 5,, = 10 pm, which are normal values for
the thermal conductivity and the thickness, the thermal resistance in the diamond will be
approximately six times that of the silicon substrate, R,, = 6R,,. Under these conditions,
the thermal conductivity of the silicon is determined instead of the diamond film. For
this reason, the silicon substrate must be etched from the backside of the sample leaving
a window of free-standing diamond. An explanation of the etching process will be given

later in Chapter 4.

3.1.1 One Dimensional Heat Conduction Model

The one dimensional heat conduction model was chosen because its solution is
readily derived. The solution is also in a form that results in quicker and easier analytical
analysis. However, the analytical advantage presents several experimental disadvantages.
One disadvantage is the extensive sample preparation that must be done to ensure one
dimensional heat flow. The top and bottom of the sample must be patterned and etched
to get a channel of free standing diamond film. The sample is also very fragile due to
the necessary geometry.

For the one dimensional case, we make the special assumption that the temperature
gradient occurs only in the x direction. This can be done experimentally by making the
characteristic length much greater than the width. With this additional assumption, the

governing heat conduction equation reduces to the partial differential equation,

62T_§ (T—T,) +g=0 3.6

1‘52? a

26
where h is the convection coefficient [W/m2 K] on both sides, 8 is the thickness of the

film, T. is the temperature of the surroundings and g is the volumetric heat generation
[W/m’] in the diamond film. A schematic diagram of the one dimensional model is

displayed in Figure 3.2.

 

 

 

 

 

 

 

 

Figure 3.2. Schematic of the one dimensional model.

The free standing film behaves as a fin suspended in air. As heat is generated in the film
a temperature distribution on the surface of the film forms about the center while the
silicon maintains a constant temperature condition at the ends.

Because equation 3.6 is of second order, two boundary conditions are required for

27

the solution. Since the temperature profile from x = 0 to x = L is assumed to be a mirror
image of the temperature from x = 0 to x = -L we can effectively describe the
temperature distribution over the entire length by modeling one half of the exposed film.
Therefore, we pick the first boundary condition at x = O. This is a boundary condition

of the second kind (Beck et al., 1992) and is described by,

61’
-k-a—J{ lat-0:0 3.7
The boundary condition at x = L is of the first kind and is described by,
T (x=L) =cons cant 3.8

The assumption of this boundary condition is made due to the silicon substrate that
remains at the edges of the sample, see Figure 3.2. The silicon which is more than 40
times the thickness of the diamond film and having a thermal conductivity of 150 W/m
K acts as a thermal heat sink. The increased thermal mass at the edges is assumed to
increase the temperature gradient across the film and allow the temperature at this
boundary to remain at a lower constant value compared to the exposed surface of the film
at steady state. Using the boundary conditions at x = 0 and x = L yields the exact one

dimensional solution,

fl] cosh(m,x) + gb +1,“ 3.9

T(X)=[(Tsi-T°')-2h cosh(mrL) 2h

 

where T,i is the temperature of the silicon at x = L and

28
[17:1, R 3.10

3.1.2 Two Dimensional Heat Conduction Model

A two dimensional heat conduction model in cartesian coordinates was chosen to
reduce some of the experimental challenges characteristic of the one dimensional model.
This two dimensional model allows for less sample preparation, heat conduction
throughout the plane of the film and a stronger film sample. Although this model is
experimentally much friendlier, it is more difficult analytically.

In the two dimensional model, the coordinate system is defined parallel to the
mutually perpendicular directions of the heat conduction so that the geometry is said to
be orthotropic. The heat conduction equation for orthotropic bodies is assumed not to
contain cross-derivatives. In addition, the thermal conductivity is considered to be the
same in the x and y directions which is known as an isotropic condition. In the two
dimensional model, heat is free to flow in the plane of the film, in directions x and y.

By applying this special assumption, the two dimensional heat equation for this case is,

kfl-I-k—afl-Eg (T—T”) +g=o 3_11

6x3 6y2 5

where the thermal conductivity is assumed to be constant throughout the film. For this
model special design considerations are also needed for the sample. Although detailed

information on the design and fabrication of the actual diamond sample is given in

29
Chapter 4, a schematic of the physical design of the sample for the two dimensional

model is displayed in Figure 3.3.

 

diamond
fllm \

etched
region

 

Side 4’ Back b

 

 

 

Figure 3.3. Schematic of the two dimensional diamond lem sample,
etched from the backside.

Here, the sample is etched from the backside resulting in a rectangular window of free
standing diamond at the center. Such a sample would still have the silicon substrate at
the edges to maintain the temperature boundary condition. By designing the sample in
this way, we can solve the governing two dimensional equation using a model which has
volumetric heat generation g [W/m’], constant temperature boundary conditions, and two

axes of symmetry. A schematic of the model is displayed in Figure 3.4.

3O

 

 

 

 

 

 

 

T-constant
I. / .
T-W diamond l
\ film |
| a
i x-O
I
l
"L I >conm
y f _.
T-constent
a y-0 b

 

 

 

Figure 3.4. Schematic of the two dimensional model.

Since the model has four symmetric quadrants, we can effectively determine the
temperature distribution on the entire face of the film by mathematically describing the
temperature distribution in just one of the quadrants. A diagram of one of the symmetric

quadrants is displayed in Figure 3.5.

31

 

T-constant
. /

l
l diamond film
q(y-O)-0 I T-eonstant

\: a /
ML

| y
o +——7————4b

th-Oi-O

 

 

 

 

Figure 3.5. One of the symmetric quadrants for the two
dimensional model.

With a boundary condition of the second kind at x = 0 and y = 0 and a boundary
condition of the fast kind at x = a and y = b, we turn to the Green’s functions, GM and

GY21 , to obtain the solution,

4 - - cos(l3mg)cos(pn%)(-1)m-1(_1),,_1
T(X’Y)="kgzz (3 13 +2"... 3.12
m-ln-l pmpn[(-f)2+(—5n)2+mf2]

 

where [in = rt(m-1/2) and [5,, = 1t(n- 1/2), (Beck et al., 1992).

32
3.1.3 Radial Heat Conduction Model

The radial heat conduction model was chosen as a compromise between the
analytical simplicity of the one dimensional model and the experimental advantages of
the two dimensional model. The steady state solution for this model is well known and
easily obtained for different conditions. Although the heat conduction would be restricted
to one direction, it would also spread throughout the plane of the film, as in the two
dimensional case.

For the one dimensional radial heat flow model, we make the special assumption
that heat conduction is restricted to the radial direction. The physical design of the
sample would be similar to the two dimensional case, but would have an exposed
diamond window which is circular instead of rectangular, see Figure 3.6 For the one

dimensional radial model the governing equation becomes,

 

dzT 1 d1" 21:
——- — = 3.13
kdzz+krdr T(T T,)+g O

A schematic drawing of the one dimensional model for radial heat flow is displayed in

Figure 3.6.

33

 

T - constant

diamond
film

 

 

 

 

Figure 3.6. Schematic of the radial heat conduction
model.

The general solution for the one dimensional radial heat flow is described by,

T(r) =ClIo(m,.z') +C2Ko (mtr) +—L +21", 3,14
mfzk

where I0 and K0 are the Bessel functions (Beck et al., 1992). Since the temperature at r

= 0 is finite, Q must be zero, and we are left with a boundary condition of the first kind,

T| r=b= cons tan t: 3.15

 

 

34
By applying this boundary condition to the general solution and solving for C1 we obtain,

T(r) =__i_ [1- 10mg)

—— +T, 3.16
mgr. I.<m.b)]

which is the solution of the temperature distribution for radial heat flow with internal heat
generation, natural convection from both front and back faces and a constant temperature
boundary condition.

Each of the three derived expressions can be thought of as a mathematical tool
which measures the thermal conductivity of a certain medium. In Chapter 4 we will
determine which of these "tools" is most sensitive in its measurement and accurate in its

estimation of the thermal conductivity of a doped diamond film.

Chapter 4

Design of the Diamond Film Sample

Perhaps the most important consideration for ensuring successful experimental
results for the estimation of the thermal conductivity of CVD diamond films is the design
of the sample. We have already encountered what adverse effects the silicon substrate
can have on the results if it is not properly etched, but there are geometrical and electrical
conditions of the sample that must also be addressed in order to optimize the experiment
and ultimately improve the results. This chapter consists of two main parts of the design
of the diamond film sample. The first part of the chapter deals with the analytical design
and consists of discussions concerning relative error reduction, heating requirements and
natural convection reduction. Out of this analytical analysis an optimum geometric and
electrical design for the sample is chosen for the experiment. The second part of the
chapter deals with the mechanical design of the sample. In this section, a diamond fihn
sample is manufactured according to the optimum parameters calculated in the first part.
Here the substrate preparation, diamond deposition, masking and etching processes are

discussed.

35

36
Analytical Design
4.1 Determining the Best Experimental Model
The first step towards designing the optimal diamond film sample is determining
which model (one dimensional, two dimensional, or radial) has the smallest uncertainty
in the estimation of the thermal conductivity. If the uncertainty of the relative
temperature rise, 0(r) (6(r) = T(r)-T.,) at a particular location r, is dominated by the

uncertainty of the thermal conductivity then,

66”) wk, 4.1

° 6k

 

where we and w, are the uncertainties in the relative temperature distribution and the
thermal conductivity, respectively (Holman, 1978). From this relationship, the relative
uncertainty in the thermal conductivity becomes,

:15: “’0 4.2
k 0(r)y(r) '

 

where

k66(r)

= 5k 4.3
y (I) —6 (I)

In this form we can concentrate on increasing the nondimensional sensitivity coefficient,
y(r), to decrease the overall relative error in the measurement of the thermal conductivity,
wk.

In the following section, 7(0) for the three different models presented in Chapter

3 is calculated and plotted as a function of the characteristic length. Parameters such as

37

film thickness and temperature rise are also varied in these calculations to show their
relative effect on the experimental error. At the end of this section, 7 for all of the
models is compared to determine which model yields the "best" estimation of the thermal

conductivity.

One Dimensional Model:

In the one dimensional model we have the relative temperature distribution,

 

= _ _ 95 cosh (mix) 95 4 4
6 (x) [ (T31 T“) 211] cosh (me) + 2h ' °
Using equation 4.4 to calculate the non-dimensional sensitivity coefficient (7) evaluated

atx=0weget

kfi
(0) = 6k = th tanh (me) 45
Y T ”‘° 2 [cosh (me) -1]

 

where

mr= __ 4.6

kb

The nondimensional sensitivity coefficient was evaluated at x=0 to determine the
sensitivity of the model to the measurement of the thermal conductivity where the
temperature rise is the greatest. We can see fiorn equations 4.5 and 4.6 that y is a
function of length (L), the thermal conductivity (k), the film thickness (5) and the natural
convection coefficient (11). In the following calculations of y, the diamond thermal

conductivity was set to k = 1000 W/m K and the convection coefficient was determined

38

from the expression,

1
(.68+.67R,‘) k,
9 4-9—
.492 '1—5 3
[1+(——P ) ]

I

15:

 

4.7

where k, is the thermal conductivity of the surrounding air and I is the characteristic
length (I = 2L) of the sample. This expression describes the average natural convection
coefficient for a vertical plate with an isothermal surface (Incropera, 1990).

To aid in the analysis of y in the one dimensional case, program ID was written.
By inputting the surface temperature of the diamond film, the surrounding temperature
of the air, the thermal conductivity of the film (k,), the film thickness and a range for the
characteristic length, 7 as a function of the total length (I) is calculated. The program
allows for temperature dependent properties of the air and prompts the user when
calibrations of these properties are extended beyond their range. The output generated
by the program is displayed in Figures 4.1 and 4.2.

In Figure 4.1, y is plotted as a function of the total length for the different film

thicknesses of 5, 10 and 15 um.

39

 

1.00-
0.95-
0.904
0.85‘
0.801

 

.7 -
81%|: ° 5
0.70-

0.55-
0.60-
0.55- ~ \ a

 

0.50-i \ -
ar - to \ q

0.00 0.01 0.02 ' 0.03 ' 0.54 ' 0.05
Length («1)

 

 

 

 

 

Figure 4.1. Nondimensional sensitivity coefficient as a function of
characteristic length for different film thicknesses, 1D case.

For each of these curves the temperature difference between the surface of the sample at
x=0 and the surroundings was set to AT = 10 °C. From Figure 4.1, we notice that the
relative error increases with increasing length and decreases with increasing film
thickness.

From equation 4.5, we see that y is also a function of h = f(AT). Figure 4.2

displays 7 as a function of length for different temperature differences.

 

1.00

 

0.95-

0.90 d

0.85 -

99M
.93.
0(0)

0.80 -

0.75 -

0.70 "l

0.65-

0.60

AT - rs

-- AT-IO
— AT-5

 

 

 

 

l
0.00 0.01

 

 

Figure 4.2. Nondimensional sensitivity coefficient vs. characteristic length
for different temperature rises, 1D case.

The film thickness was held at 10 pm while the temperature differences were set to 5, 10
and 15 °C. Although at a given length the graph displays an decrease in y with an
increase in the temperature difference, we see that the relative error of the thermal

conductivity is more sensitive to the film thickness than the relative temperature rise when

comparing Figures 4.1 and 4.2.

 

41
Two Dimensional Model:

In the two dimensional model we have the relative temperature distribution,

COS (Bm—a’f)cos (911%) (-1)m-1(_1)n-1

 

6(XIY) =£€22 - - 4.8
kmln-l Bp [(££)2+(&)2+m2]
m n a b t
Evaluating this expression at x = O and y = 0,
y(0.0)=£2 [1- 1 4.9

 

m=1n=1 [(%)2+(%)2+mtz]

The nondimensional sensitivity coefficient is determined at (0,0) because it is at this
location where the relative temperature rise is at its peak and the radiometer’s signal-to-
noise ratio is maximized. We can see from equation 4.9 that 7 is a function of the
characteristic length (a), the characteristic width (b) and m,. The fin term, m,, is
dependent on the thermal conductivity (k), film thickness (6) and the natural convection
coefficient (h), (refer to equation 4.6). To aid in the analysis of 'y in the two dimensional
case, program 2D was written. After inputting the surface and surrounding temperatures,
the thermal conductivity, the film thickness and a range for the characteristic length (l),
where l = 2a and the length and width are equal, 7 is calculated for the two dimensional
case as a function of the total length. The program also allows for temperature dependent
properties of air.

By calculating 7 as a function of length for different film thicknesses, we see the

42

same behavior as in the one dimensional case.

 

 

 

 

 

 

: ‘d's .
0.75- \ o _

: "a ‘

‘ it '1000‘0 w/ x ‘

a - \
0.70 . ,m . , . , . , .

0.00 0.01 0.02 0.03 0.04 0.05

Length (m) - Width

 

 

 

Figure 4.3. Nondimensional sensitivity coefficient vs. characteristic length for
different film thicknesses, 2D case.

Again, the relative error of the estimated thermal conductivity increases with increasing
length and decreases with increasing thickness. Figure 4.3 also shows that increasing 7
by increasing the film thickness becomes more challenging for thicker samples. By
increasing the thickness fiom 5 pm to 10 pm at a length of 4 cm, we increase 7 by 0.12.
However, if the film thickness is increased from 20 pm to 30 pm, a 10 pm difi‘erence, we
only increase the value of y by a mere 0.03. This information is beneficial in designing

an optimal experimental sample having a relatively thin diamond film while keeping

43

diamond deposition time to a minimum.
The 7 values here corresponding to the same film thickness and AT are greater
than the one dimensional case. A comparison of y for both the one and two dimensional

models is presented in Figure 4.4.

 

 

  
 

1111111

20. O(x.y)

1 141411111

1 0-10urn
:AT-IO
0.60 k-1WWImK r 1

r r r ' T
0.00 0.01 0.02 0.03 0.04 0.05
Length (m)

111411111 llll

 

 

 

 

 

 

Figure 4.4. Comparison of the nondimensional sensitivity coefficient for the one
dimensional and two dimensional cases at the same characteristic length.

We can see from Figure 4.4 that the relative error in the estimation of the thermal
conductivity is consistently higher in the one dimensional experiment than in the two
dimensional experiment. At I = 5 cm and 8 = 10 pm, “0.0) for the two dimensional
model is approximately 0.76 where 7(0) for the one dimensional model corresponding to

the same length is only 0.64.

44
Figure 4.5 is a plot of 7 as a function of length for different temperature

differences in the two dimensional case.

 

 

 

 

 

 

0.75:
: --- LT '- 10
-" AT - 10 ‘ - 1°
j—AT-O tt-tOO‘t‘ITV/mit
0.70 I 1 t I - j 1 i I I’
0.00 0.01 0.02 0.03 0.04 0.05

Length (m) - Width

 

 

 

Figure 4.5. Nondimensional sensitivity coefficient vs. characteristic length for
different temperature rises, 2D case.

We can see here that 7, as in the one dimensional model, is not as sensitive to the
temperature rise as compared to a change in the film thickness. In fact, for lengths of
approximately .01 m or less the change in 7 with respect to the steady state temperature
difference is almost negligible.

Since the analysis has dealt with a square sample, it is interesting to determine if
there is a better rectangular geometry that would improve the estimates of k or if there

is a geometrical arrangement we should avoid altogether. Figure 4.6 is a plot of 7 as a

45

function of the characteristic length for different sample widths (1.0, 0.5 and 0.25 cm).

 

 

 

 

 

 

1.000 ‘ I I v I t l ' r 1
1 V ,
‘ «1
0.0054 '\ ----- =
1 ________________ ,
e \ ___________ 1
0.9904; '. _-
3 g i ' -
at . ‘ .4
SJQIO 0.985.: ‘. -:

q \
0.980- ’ . ..
0.975% --------- J
1 O-tolun --u-1.0crn:
1 “-10 -- w-.50crn.
0.970 . r , r k .' IOOO'W/m K1 —' w - :25 cm‘
0.00 0.01 0.02 0.03 0.04 0.135
Length (111)

 

 

 

Figure 4.6. Nondimensional sensitivity coefficient vs. characteristic length for
different sample widths of 1.0, 0.50 and 0.25 cm, 2D case.

By holding the widths constant we see that all three curves have a minimum value for
7(0,0) where the total length equals twice the total width (az2b). Although all of the
values for 7 are above 0.975 for 5 = 10 pm, AT = 10 and k = 1000 W/m K, this
geometrical ratio should probably be avoided when designing a diamond film sample for
this two dimensional model. For it is this ratio, a/b=2, where the relative error in the

estimation of the thermal conductivity is greatest

Radial Model:
In the one dimensional radial model the relative temperature distribution is
described by the expression

0(1) =_L [1_ Io(mfr)

mgk —Io (mfb) ] 4.10

Evaluating this expression at r = 0,

 

m‘bt Ilmfb) 4.11

Y(0)= 2 Io(mgb)2-Io(mfb)]

where b is the radius of the sample. As in the one dimensional and two dimensional
analyses, a computer program was written for calculating 7(0) for the radial case.
Program RAD was written using the Student version of Matlab. Matlab was used in this
case due to its ability to quickly and easily evaluate I, and Il Bessel functions. RAD uses
the same input values of film thickness, steady state temperature difference, and thenrral
conductivity as in the one and two dimensional cases, but prompts the user for the
diameter of the sample which is the characteristic length.

In Figure 4.7, 7(0) is again plotted as a function of the characteristic length

(diameter) for different film thicknesses.

 

47

 

 

 

 

 

 

A'l' - ro .
0.75 " ' ‘.°°° "4'“ " . . . , . , .
0.00 0.01 0.02 0.03 0.04 0.05

Diameter (m)

 

 

 

Figure 4.7. Nondimensional sensitivity coefficient vs. diameter for different
film thicknesses, radial case.

We can see from the results in Figure 4.7 that 7(0) behaves in much the same way as in
the one and two dimensional cases. Values for 7(0) decrease with increasing diameter

but increase with increasing sample thickness.

48

 

 

 

0.85d —
j — 41' - 5 7
." 3'3 o-roum I
0.80 k'- 1000 “If" K

 

 

 

0.00 ' 0.01 0.02 V 0.03 ' 0.04 ' 0.05
Diameter (m)

 

 

 

Figure 4.8. Nondimensional sensitivity coefficient vs. diameter for different
temperature rises, radial case.

Figure 4.8 shows again that even for the radial case that 7(0) is more sensitive to the film
thickness than the steady state temperature difference, AT. The temperature difference
again has little effect on the relative error of the experiment for diameters of
approximately .01 m or less.

In order to choose the experimental design of the sample which would most limit
the relative error of the estimation of the thermal conductivity, 7(0) versus the
characteristic length for all three cases (one dimensional, two dimensional and radial) is
compared at a film thickness of 10 um, a AT of 10 °C and a thermal conductivity of 1000

W/m K. These parameter values were chosen to simulate actual experimental conditions.

49

The plot for all three cases can be seen in Figure 4.9.

 

 

  

0
It

0

O

In

I

I

I

11111114 11 1 111111111 1441111111

6-10prn
41-10

0.5;)-L'm/m K , .

0.00 0.01 0.02 0.03 0.04
Characteristic Length (m)

1111

 

 

 

.0
G
or

 

 

 

Figure 4.9. Comparison of the nondimensional sensitivity coefficient vs.
characteristic length for 1D, 2D and radial cases.

We can see that 7(0) for the radial model is consistently higher than both the one
dimensional case and the two dimensional case. From this analysis using the
nondimensionalized sensitivity coefficient, 7(0), we have determined that the radial model
will produce the most accurate experiment for the determination of the thermal

conductivity.

50
4.2 Special Requirements for the Desired Temperature Rise

Now that we have determined the "best" analytical model for the experiment, we
must deal with some special heating requirements for the diamond film. Due to
diamond’s unique semiconduction capabilities (in its doped state), we are able to use
electrical resistive heating within the doped film to create the necessary temperature
gradient. However, in using internal heat generation within the film we must be certain
that 1) there is volumetric heat generation in only the diamond film region and 2) the

electrical resistivity of the film is low enough so the film can be heated with the available

power supply.

4.2.1 Volumetric Heat Generation in Diamond

Since diamond is deposited on a silicon substrate during the manufacturing
process, we must be certain that as we try to heat the diamond film the electrical current
does not leak into the silicon which also conducts electricity. If there is not electrical
insulation between the two semiconducting layers, substantial error in quantifying the
volumetric heat generation within the diamond film will occur. If the electrical leakage
is substantial, there could even be volumetric heat generation within the silicon itself,
completely invalidating the experimental results.

To combat this problem, a thin layer of SiO2 is commonly deposited on top of the
silicon substrate. The SiOz, an electrical nonconductor, acts as an insulating medium
between the deposited diamond and the silicon substrate. A diagram of a sample with the

additional layer of SiO2 is displayed in Figure 4.10.

51

 

doped diamond

 

 

 

 

Figure 4.10. Diagram of sample with a doped diamond film, a layer of
$102 and a silicon substrate.

However, this substrate alone cannot always provide adequate electrical insulation when
the silicon is etched from the backside. The etchant chemical which is used to dissolve
the silicon can settle between the silicon and diamond layers destroying the thin SiO2
layer. So, in order to ensure electrical insulation, a thin layer of undoped (electrically
nonconductive) diamond is deposited first on top of the 8102 layer. A diagram of this

type of sample can be seen in Figure 4.11.

 

 

 

 

 

Figure 4.11. Schematic of sample with doped and undoped
diamond films, a layer of SiO2 and a silicon substrate.

 

 

52
Since diamond is impermeable to the etchant, the undoped layer provides a solid

electrically nonconductive medium between the doped diamond region and the silicon
substrate. We would like to_have the undoped layer as thin as possible (a 1 pm) because
this layer can limit our ability to heat the doped diamond film and it may have different

thermal properties.

4.2.2 Doping Requirements for the Diamond

In order to allow for the internal heat generation necessary to heat the diamond,
we must dope the diamond film with boron to obtain a certain electrical resistivity. The
electrical resistivity is inversely proportional to the level of boron added before the
diamond deposition process. If the resistivity is too high then the available power supply
(120 volts, 1.5 amps) will be insufficient in generating the necessary power to heat the
film. In addition, the resistivity must be low enough to compensate for the heat sink
behavior that the undoped diamond layer will have. We need to avoid a situation in
which the undoped layer of diamond dissipates all of the heat generated in the doped
layer and creates a uniform temperature distribution across the surface of the film.

Currently, the Microstructures Laboratory which makes these diamond samples can
manufacture films having a resistivity (p) of 0.5 Q-cm. To see if this resistivity is

sufficient to heat the diamond film we turn to the revised solution of the radial model

960! I 0(meft'r)
0(r)= [1- I :(mefl‘b)] 4.12

where

 

53

 

= 211 4.13

and 8d and 8,, are the thicknesses of the doped and undoped diamond layers, respectively.
Evaluating this solution at r = 0, we find the relative temperature rise at the center of the

exposed diamond to be

 

-99d _ 1
0(0) ——271_ [1 10(m6ftb) ] 4.14

The volumetric heat generation (g) is equivalent to electrical power per unit volume and

is calculated from the expression,

9: V” 4.15
(12* Vol ume)
where V is the applied voltage and R is the electrical resistance. To obtain uniform heat

generation we use the electrical circuit displayed in Figure 4.12 where V and L

correspond to the applied voltage and the length of the circuit, respectively.

 

 

 

 

dopeddiamondtllm
,. we“
// \v/
\ E v
L \ / __
\\.//

 

 

 

 

 

 

 

 

 

Figure 4.12. Schematic of the electrical circuit.

54

The electrical resistance in this case is defined as,

R=££ , 4.16
AC
where p is the resistivity of the film and Ac is the cross sectional area of the doped film.
Substituting equations 4.15 and 4.16 into equation 4.14, we develop the relationship
between the applied voltage, the electrical resistivity of the film and the relative
temperature rise to be,
1 1

I O (meffb) 4.17
2hpL2

Vzbd[l-

 

 

where L is the distance the current travels in the circuit (L 2 diameter of the etched
region). This relationship is useful in pre-determining the temperature range of the
experiment according to the sample’s geometry, resistivity and the available electrical
power. The relative temperature rise, 0, is calculated as a function of voltage with the
resistivity set to 0.5 Q-cm, a circuit length of 2 cm, and the diameter of the etched region

fixed at 1 cm. The results are presented in Figure 4.13.

55

 

 

 

 

 

 

4o 7 I U I U l I I U I T '7’
-- 6(undoped) - 2 um I’
‘ — 6(urtdopod) - 1 pm ’1
30d
'3 20-
10-
.
0
0
Voltage (volts)

 

 

 

Figure 4.13. Temperature rise as a function of applied voltage for different
film thicknesses.

The four curves in Figure 4.13 represent the voltage requirements for heating films with
doped thicknesses of 5 and 10 pm and undoped thicknesses of 1 and 2 pm. Although
limited to 120 dc volts for the applied voltage, this prediction indicates that we can
achieve approximately a 40 °C temperature rise with a sample having a doped thickness
of 10 pm and a thermal conductivity of 1000 W/m K. Likewise, a 30 °C temperature rise
can be reached having doped and undoped film thicknesses of 5 and 1 pm, respectively.
It is interesting to note that the heating requirements for the 8,1 = 10 pm, 8,, = 2 pm and
the 8d = 5 pm and 8“ = 1 pm cases are nearly the same. This is to be expected since the

ratio of the 2 layers in each case is 5:1. Therefore, if the highest doping level is used,

56

a Hall concentration of 9.6 x 10" cm'3 resulting in a resistivity of 0.5 Q-cm at 300 K, the
heating requirements for the doped diamond film should be adequate to achieve the goals

of the experiment, even with the addition of the undoped layer of diamond.

4.3 Limiting Natural Convection

In most of the previous works dealing with the measurement of the thermal
properties of CVD diamond films listed in Chapter 2, natural convection was addressed
as a source of uncertainty in the experimental setup. Because natural convection is
another form of heat transfer (other than conduction), it can pose serious problems to the
estimation of the thermal conductivity if it is not limited Figure 4.14 is a plot of 7(0)
versus the product of the fin term m, and the characteristic length for the tlu'ee models
discussed in Chapter 3. This product is a nondimensional parameter that relates to the

amount of heat lost by convection.

57

 

   
 

 

l1111l1111l1111

111111111l1111

  

0 55; — radial model “
' : — - two dimensional model ‘
1 -- one dimensional model

I r r r r
0.0 0.5 1.0 1.5 2.0 2.5

O
at
O
14
I
I
f 1 1111114111l{11

 

 

 

9'
o

m o characteristic length

I

 

 

 

Figure 4.14. Nondimensional sensitivity coefficient vs. (m,*characteristic
length) for the 1D, 2D, and radial cases.

We can see from Figure 4.14 that as convection becomes more dominant the relative error
in the thermal conductivity increases. Not only can the error in the estimated thermal
conductivity be high but errors in the calculation of the convection coefficient can also
be significant.

Since the natural convection coefficient must be calculated from position and
temperature dependent empirical expressions, values tend to represent approximations

rather than exact quantities. In light of this, some engineers and physicists have designed

58

vacuum chambers in an attempt to negate the effects of natural convection. Theoretically,
a perfect vacuum would indeed negate convection. However, practically, a vacuum
chamber presents additional experimental challenges; one of which is the design of the
chamber itself. Additional instruments, measurements and seals around electrical lead
wires and other openings only add to the experimental difficulty. In many cases the
vacuum chamber can prove to be more of an experimental inconvenience rather than an
aid.

Since good vacuum chambers are either time consuming to build or expensive to
buy, one of the goals of the present work was to design an experimental setup which was
simple, effective and did not require the use of a vacuum chamber to minimize natural
convection. In order to do this, however, we must properly choose the geometry of the
sample. We see from Figure 4.14 that by decreasing the characteristic length we decrease
natural convection. Recall that the convection heat transfer is proportional to the surface
area. By comparing the surface areas for the two dimensional and the radial cases, we
see that the characteristic lengths are related by n/4 D2 = L2, where L is the length of the
two dimensional sample and D is the diameter of the radial sample. Since the
characteristic lengths are set equal in Figure 4.14 the radial curve and the two dimensional
curve become separated by a factor of (rt/4M ~ 0.9.

In order to quantify the relative amount of heat lost by natural convection
compared to the volumetric heat generation we again turn to the mathematical model for

the radial case

 

 

0%) 2” e 954 -o 4.13

’ k(8d+8u) * mega.) ’

$.61
IdI

59
which now includes the thin layer of undoped diamond (8“); the solution is described by

equation 4.12. If we divide the governing differential equation, equation 4.18, by
[g8d/k(8d+5.)]. the nondimensional convection term becomes [-2h0/g8d]. This parameter
represents the percentage of the total heat generated which is lost to natural convection.

Substituting equation 4.14 into the nondimensional convective term, [-2h0/g8,,], we get

 

2220(0) =1_ 1 419
I 90d I Io(meffb) .

We again evaluate 0 at 1:0 since it is at this center location where the temperature rise

is greatest. A plot of this parameter as a function of the sample diameter at different film

thicknesses is presented in Figure 4.15.

 

 

 

 

 

 

0.10 V l U 1 f O I
-- 6(undopod) - r um I,’ ,4
0.09- — 6(undopodl-2nm f; I’
0.08- ’ -
0.07- a
0.06- -
° .
r5113 005- ~
0.04- -
0.03- ..
0.02-n ~
0.01 .. -
0.00“ I '4 r I v I 1
0.00 0.01 0.02 0.03 0.04
Diameter (m)

 

 

 

Figure 4.15. Ratio of natural convection to the total heat generated in
doped film vs. diameter for different film thicknesses.

60

We can see from Figure 4.15 that with 5,, = 10 um, convection contributes approximately
1% of the total heat dissipated assuming a sample diameter of 1 cm and a thermal
conductivity of 1000 W/m K. Likewise, when 8,I = 5 pm convection contributes just
under 2% of the heat loss. In addition, we also see that the undoped layer of diamond
decreases the effects of natural convection.

Before choosing a geometry for the sample, we need to once again calculate 1(0)
as a function of the diameter for different film thicknesses (mainly 10 and 5 pm) to see
the effects the undoped diamond layer has on the relative error. The nondimensional
sensitivity coefficient for doped thicknesses of 10 pm and 5 pm and undoped thicknesses

of 1 pm and 2 pm as a function of diameter is presented in Figure 4.16.

 

 

 

:—undopedloyer-2nm u-rooo
080 --undopedlgyer-1pm 51-10

0.00 0.51 0.52 0.53 ' 0.54 ' 0.05

Diameter (m)

 

 

 

 

 

 

Figure 4.16. Nondimensional sensitivity coefficient vs. diameter for
different doped and undoped film thicknesses.

 

61

Although the undoped layer of diamond decreases our relative temperature rise, we see
from the results of Figure 4.16 that the undoped layer of diamond can also help decrease
the error in the estimation of the thermal conductivity. However if the thickness of the
undoped film is relatively large compared to the doped film the estimated thermal
conductivity will be indicative of the undoped region not the doped, a situation that we

want to avoid.

4.4 Analytical Results

From the results of the analytical analysis presented in this section we can choose
specific geometrical and electrical parameters for the design of the diamond sample that
will optimize the radial experiment.

A diameter of approximately 1 cm is a desirable characteristic length because the
relative error is little-affected by both the relative temperature rise and the thickness of
the doped film in this region. This diameter is also advantageous since the infrared
scanner will have no problem imaging an object of this size.

A Hall concentration of 9.6 x 1018 cm“3 resulting in the resistivity (p) of 0.5 Q-cm
is a suitable doping level to create a temperature rise of 30-40 °C, depending upon the
actual thicknesses of the doped and undoped layers. It is recommended that the resistivity
not exceed 1 Q—cm for film thicknesses on the order of 10 pm if using a 120 v, 1.5 Amp
power supply. For higher film resistivities, the power supply may be unable to heat the
sample.

Although the results become better as the thickness of the doped layer increases,

reasonable values for 8,, range from 5-10 pm. This is due to the slow deposition rate of

62
the current hot-filament deposition chamber in the Microstructures Laboratory. However,

in this range, convection adds up to only 2% of the total heat dissipated for the worst case
(8d = 5 pm, 8,, = 1 pm ) and the nondimensional sensitivity coefficient, 7(0), is at a high
value of 0.985.

An undoped diamond layer between 1-2 pm is also suggested by the analysis.
This is the range at which the film surface becomes continuous. A continuous layer is
essential in ensuring electrical insulation between the two semiconducting media. The
undoped layer of diamond should not only allow for an adequate temperature rise but

should also help reduce the relative error and the natural convection in the experiment.

diameter of exposed diamond ‘ S 1 cm
thickness of doped diamond layer (6,.) ‘ 5-10 pm
thickness of undoped diamond layer (5,.) 1-2 pm

 

 

 

 

resistivity of doped region (p) ' s 1.0 Q-cm

Table 4.1. Optimum sample design parameters

These results are determined from assuming a thermal conductivity of 1000 W/m K.

Mechanical Design
This section describes specialized design techniques utilized by the Microstructures
Laboratory at Michigan State University’s Research Complex to fabricate diamond

samples described by the results of the preceding analytical analysis. The design of the

63

diamond sample itself consists of two main parts, the topside fabrication of the doped and

undoped layers of diamond and the bottomside fabrication of the patterned diamond mask.

4.5 Topside Fabrication

Before diamond can be deposited on either side of the substrate, the supporting
silicon wafer must be carefully prepared and cleaned The wafer, approximately 450 pm
thick, with a thin layer of SiO2 on top is first cut to a 2 x 2 cm2 square. The square is
cleaned thoroughly using pure acetone and then methanol, rinsed with distilled water and
then dried with nitrogen gas.

With the wafer properly cut and cleaned, a solution known as photoresist is
applied to the top surface of the $0,. This solution is applied by spinning the wafer in
a rotation machine. If the surface of the wafer is free of debris then photoresist can be
evenly distributed across the surface. The sample is then placed in an oven set at a
temperature of about 400 °C for approximately 30 min. in order for the photoresist to
adhere to the $0,. The application of the photoresist is an important factor in the
success of the diamond deposition since it is this solution in which the diamond powder
and other elements necessary to grow the diamond crystals are present.

Upon completion of the substrate preparation, the diamond film fabrication process
for the topside is commenced. The sample is placed in a container known as the
deposition chamber. This chamber contains the electrical filaments which heat the sample
and the necessary carbon-enriched gas mixtures for deposition. A picture of the

deposition chamber is shown in Figure 4.17.

    

Figure 4.17. A photograph of the hot
filament CVD diamond deposition chamber.

The sample rests inside the deposition chamber on a platform several inches below the
heating filaments. The chamber is then closed and vacuum is applied so that the quantity
of the gases can be carefully regulated once the process is activated. With the gases
pumped out of the chamber, the power is switched on such that there is approximately
4 mA of current flowing through each of the six filaments which are connected in
parallel. As the filaments begin to heat up the surface of the sample, hydrogen gas is
injected into the chamber for approximately 3 min so the surface of the photoresist can
be cleaned. When the pressure is less than 10 mtorr in the deposition chamber, the gas

valves controlling the flow of the carbon enriched gases (CH4, CIH2 and C0) are then

65
opened. As the filament temperature reaches 2000 °C, the carbon from the injected gases

begins to deposit on the crystals already present in the photoresist. As the carbon crystals
grow in size and begin to fill in throughout the surface of the substrate, a continuous layer
of diamond eventually results. Due to the grain size of the crystals and the fill-in pattern
on the surface of the substrate, the film becomes continuous between thicknesses of l and
2 pm. With a growth rate of only 0.25 um/hr, the process of fabricating the undoped
layer of diamond lasts approximately four hours.

The thickness of the undoped layer for both samples was measured with a
scanning electron microscope (SEM). Measurements were taken along the plane parallel
to the film surface to determine the thickness of this layer. In the graphs generated by
the SEM, Figures 4.18-4.2l, the x axis is in units of pm and the y axis is in angstroms

(A), 10'10 m

 

4404 an: 2mm near: 94 a
4 06~1r~93 seeeo LOH HOHIZ= i3 Bzgufl

45.060
43.000
35 .- 993
33 a 330
25 .- (383
26,900
15,886
rareaa
5 .- 9:103
a

use-r
#0
Sub

 

“N ' Ft cunsda =- mar
Bu" SLonH DEKTnK II

 

 

 

Figure 4.18. Thickness of the undoped diamond film on sample A measured
in A by the SEM.

We see from this measurement that the undoped film on sample A has a continuous layer
approximately 10,000 A (1.0 pm) thick. The large peak near the center of the graph is

due to small debris, probably dust particles.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

igiggdagsrrr-aa 3352 iufigfl ,_, H3E?£§ iigggpr
' r 3 - ' M.aimfirs.aae
i m g _ 2 '14.808
;-'§ - ’Efrira.eoc
g l -- l W W E, ’ 16; race
_a_~_arépflfi I‘Lfiéflg 8.000
M. 4: - - a _ E 5.- 339
”4-15% - 1 g 4.666
: - "__ w w = 2.000
(1')] E J A M w i“ E B
‘- ‘ ‘t‘ m . . 1 .. W‘s-.326 ace
cum: 6.?83 a a 4430" R CURSOR " 443
M cue: 3.390 n a 1.8?2em SLOHN DEKTAK II

 

 

Figure 4.19. Thickness of the undoped diamond film on sample B measured in A by
the SEM.

Figure 4.19 is a measurement of the undoped layer of sample B where the thickness of
the undoped layer of diamond is approximately 8,000 A (0.8 pm).

With the undoped layer approximately 1pm thick for each sample and by checking

 

67

the electrical resistance of the films we can conclude that the undoped diamond films are
sufficient in providing the necessary electrical insulation from the silicon substrate.

With the first layer completed, the process is stopped and preparations begin to
fabricate the doped diamond layer. After the sample has cooled, a boron holder is
subsequently placed on the surface of the film, directly under the filaments. The holder
is in a honeycomb configuration, housing many openings where the boron powder is
added. By filling all of the holes in this holder, the highest available doping level for the
doped layer is achieved. After the holder is filled, the deposition process is again
repeated. However, in order to ensure uniform doping throughout the film, the film is
redoped for every 4 pm of growth.

After approximately 10 hours of deposition, the process is terminated and the
thicknesses are again measured using the SEM. Figures 4.20 and 4.21 are SEM

measurements of the doped films for both samples.

68

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

. ~ >2" . at . ‘ t ‘ . .
. . z-x. , ~ A. ~ .
" l ' _ 4:. -I~..o:. ;. A ' ' ‘
' ' I .- ' - ‘ ' I. v . r l
> e e v t. I . . . .
, » -: .\ .‘ ‘ : .~'..” '. ‘. . t
' ’3 . .- - .. a ‘ ’ l . .
. .
. . , w . ,
'.. ‘ . . ' .‘ .. ' ‘ .1”, ' .. .. J
'.. ¢_ . '1; .r. ~ .- ..
_ , . . . .z,
. ' r e "'. . 4 '\ .. '.- t ‘ .. ..
V , . ‘ ‘ .'.'\'.. ”r. . . -
.. fl 3 ~ ‘ ‘2 '
1 ‘ ‘ ‘ . ‘ ..'.
"I not. .24 a, _: . . i- 4
a . . ,. ~ "r. ‘I
. . . .. .- .x-..- . . .. . - . . ,. :~.
.n . . s p ., u ,. ..
' ' ~'..\' A; ‘ ‘: 'c‘ -. is;
' '-'.\':’= :v’..- 3" 1' - .' ' . ~. ‘ -:- .-:;r
. .M. .' A'.’ . . ' -,. .p -
.' u . . -\\l/ \ ’ '3 ‘ ' \ ' " '.
-, ." ~ . - . w -. . ' v .
I _ . . . . r . .
A , J , . _ . . .. k
. - _ . ' u '-
J‘; 1. .. w \' . . A . ' x a
.'. ...¢ . A v 'v, _“.- -_,'_ ' ._ . '
.... . :4 i - . .l‘ \ 1
- .- . ; . .~ - :~
. . a . .4‘. . . .

 

 

El 2:!" ' "t :10". , cl": = . , ".1 . ,‘.
n, can: sneeze at 972m 906' HT" 56191.5 a
P1 CUH= 55:96.0 R Ii 1’992ul’1 'SLOFIN DEKTRK II
Figure 4.20. Thickness of the doped film on sample A measured in A by the
SEM.

 

 

 

 

 

 

Figure 4.20 shows a thickness of approximately 56,000 A (5.6 pm) for the doped diamond
layer on sample A, while Figure 4.21 shows an increasing thickness of nearly 60,000 A

(6 pm) for the doped region on sample B.

69

 

 

at.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.3a: q '
H. :2: tamer
he” ._ieaiffl%p
it i ' Xvi") ‘:':-."_' 3198’" ’“w‘fi
1%3'.’ 5} "' ? i-ee.m
7'23: ' 23m
2 ‘0
5.: (20.909
a cum 04.816 9 a 7.3230" ‘ ‘ "HUG-’HT- B34:?9.:':9..
l‘1 CURz 87.19? A (I B. 933.0" . SLOGN DEKTfiK I!
Figure 4.21. Thickness of the doped diamond film on sample B measured in

A by the SEM.

4.6 Bottomside Sample Fabrication

When designing a sample that has a free standing circular diamond window 5 1
cm in diameter, specialized fabrication techniques must be used on the bottom of the
sample. For instance, in order to remove the silicon backing in the area of interest and
maintain the sample’s structural integrity by retaining silicon in other areas, the diamond
must first be patterned on the backside. Patterning, commonly known as masking, is a
technique which enables diamond to be deposited and grown in precise surface

configurations.

To pattern the samples, a mask is first designed. The mask is used to block UV

70
rays which bombard the photoresist after it has adhered to the back of the sample. By

exposing the photoresist to UV radiation, diamond growth occurs in the regions shielded
by the mask and is eliminated in the exposed regions.

After the back has been patterned, the undoped diamond deposition process is
again repeated. Unlike the top layers of diamond, the thiclmess of the bottom layer has
no significant effects on the results. However, the layer needs to be continuous to guard
against excessive removal of the silicon during etching.

Pictures of one of the diamond samples made by the Microstructures Fabrication

Laboratory is displayed in Figures 4.22 and 4.23.

 

    

 

 

Figure 4.22. Deposited doped diamond
film on the front side of sample A.

71

 

 

 

 

Figure 4.23. Diamond mask patterned on
the back side of sample A.

With the diamond deposition completed on the backside, the sample can now be etched.
To etch the sample a solution consisting of 25 ml of hydrofluoric acid (HF), 50 ml of
nitric acid (HNO3), and 25 ml of acetic acid (CH3COOH) is prepared. The solution is
then carefully deposited (using a micro pipette) inside the circular region on the backside
of the sample where it attacks the exposed silicon. Since the solution is applied to the
silicon substrate in small amounts, the acid and its potency tend to evaporate rather
quickly. In light of this, the solution is periodically deposited to the silicon surface until
it has penetrated through to the undoped layer of diamond on the topside of the sample.

This process, however, is a tedious one and takes several days to complete. With the

72

silicon completely removed in the circular region inside the diamond mask, the top layers

of diamond are now free standing and exposed on both front and back sides.

Chapter 5

Experimental Techniques

Experimentally, diamond films are one of the most challenging materials to
analyze thermally. Their microstructural size and rapid thermal response pose both spatial
and temporal problems for experimental techniques utilizing surface mounted temperature
sensors such as thermocouples and resistance thermometers. For this reason an optical,
non-contact technique was implemented for temperature acquisition. This technique
detects the thermal radiation emitted by an object and calculates the temperature.

The experimental techniques developed for the steady state temperature
measurements of doped diamond films are described in this chapter. Included are

subsections detailing the data-acquisition system and the experimental setup.

5.1 The Data Acquisition System

The data acquisition system is comprised of three primary components: 1) the
Model 600L Infrared Imaging Radiometer, 2) the Thermal Image Processing System and
3) a digital multimeter. The first two components, the infrared radiometer and the image

processor, comprised the temperature measurement system, while the digital multimeter

73

74

is used to measure the power generated in the electrical circuit. A diagram of the data

acquisition system is displayed in Figure 5.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lieutenant-d
00mm
WW ¥ I
--:I=- l l -
out-rum
Germanium
terrain-W
— mm mm

 

 

 

 

 

 

 

 

 

 

 

Figure 5.1. Diagram of the data acquisition system.

5.1.1 The Inframetrics Model 600L Infrared Imaging Radiometer

The Inframetrics Model 600L Infrared Imaging Radiometer, consisting of an
infrared scanner and an electronic control unit, is designed for applications requiring 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 and background correction. The system can also be linked with a

video cassette recorder or a thermal image processor to store thermal events. The Model

75

600L IR Radiometer complete with scanner, RGB monitor, control unit and VCR is

displayed in Figure 5.2.

 

Figure 5.2. The Inframetrics 600L IR Radiometer.

The scanner incorporates two independent electromechanical servos
(galvanometers) which perform horizontal and vertical scanning. Attached to these servos
are scanning mirrors which are contained in a sealed, evacuated module for increased
efficiency. Horizontal scanning is performed at the very high rate of 8 kHz in a resonant
sinusoidal mode. Vertical scanning is performed in a sawtooth pattern consistent with
standard TV formats at 60 Hz.

The thermal radiation entering the scanner through a germanium window is

76

deflected by the horizontal and vertical scanning mirrors and is focused on a HngTe
detector. Motorized focus and zoom mechanisms are operated within the scanner by
remote control. With the HngTe detector cooled by liquid nitrogen to approximately
77 K, maximum thermal sensitivity is obtained.

The signal generated by the HngTe detector is processed, digitized, reformatted
and prepared for visual display by the control unit. A microprocessor within this unit
accesses such functions as background temperature, temperature range and emittance
settings, image averaging, fast line scanning, focusing and zooming. The microprocessor
also performs internal calibrations as the temperature of the scanner changes and when
filters or lenses are connected to the scanner. The control unit accesses individual picture
elements, then calculates temperatures according to calibration curves which were
measured at the factory and stored in a read-only-memory. Due to variations to spectral
response, every system has a unique calibration curve. The output from the control unit
is a standard RS-170 format with 8 bit resolution, providing 256 pixels/line.

In order to increase the spatial resolution when imaging smaller objects, two
additional lenses are installed on the front of the scanner. One lens in a 3x telescopic
lens which is connected directly to the scanner. Attached to the telescopic lens, however,
is a 6" closeup lens which can resolve objects smaller than 100 pm. A photograph of the

two lenses is displayed in Figure 5.3.

77

  

6' close-up lens

3x telescopic lens
Figure 5.3. The 6" close-up lens and the 3x

telescopic lens used to increase the spatial
resolution of the radiometer.

5.1.2 Thermal Image Processing System

The Thermal Image Processing System is primarily used to capture and store
entire thermal patterns and access temperatures corresponding to any or all digitized
picture elements (pixels) which make up the thermal field. Each field consists of 365 x
280 pixels, totaling over 102,000 accessible surface temperature measurements.

Temperature measurements are individually calculated from the radiant intensity

78
of each pixel, which is digitized and saved in an 8 bit format. When the minimal 5 °C

temperature range is activated on the control unit of the Radiometer, a maximum thermal
resolution of 0.03 °C is attained.

By using the external optics previously discussed and seen in Figure 5.4 and by
zooming in at the minimum field of view allowed by the radiometer, the spatial resolution
shrinks to approximately 1 pixel / 10 pm. With this type of spatial resolution, the
smallest area that can be fully imaged is approximately 4 mm x 3 mm.

The Thermal Image Processing System resides in an Epson 286, 16 M2 PC. Since
the computer is only used to run the driver software for the image processor, access its

functions and store images, the 286 is an adequate machine.

5.1.3 The Omega 880 Digital Multimeter

In order to simplify the experimental setup and resulting experimental procedures,
the Omega 880 Digital Multimeter is used to measure the power supplied to the electrical
circuit to calculate the internal heat generation within the doped diamond film region.

A picture of the multimeter is displayed in Figure 5.4.

79

 

altimeter.

The multimeter is used to check and measure the applied voltage and determine the
current flowing through the circuit usually by measuring the voltage across a known

resistance, see Figure 5.1.

5.2 System Calibration Check

Since the radiometer uses inaccessible internal calibration curves which were
measured by the supplier to calculate temperature, it is important that the performance of
these tables be calibrated versus a known temperature in order to validate this temperature

measurement system. The radiometer was calibrated using a vertically mounted "black"

80
aluminum plate which was uniformly heated from the backside and had a thermocouple

attached to the center of the frontside. The thermocouple was used to measure the
temperature of the plate. Assuming the emissivity of the plate to be 0.98, the temperature
measured by the radiometer was the same as the temperature measmed by the
thermocouple. In this case, the emissivity that is input into the control unit of the
radiometer merely acts as a calibration factor which initializes the "actual" temperature
of the thermocouple and measured temperature of the radiometer.

As the temperature of the plate increased, the corresponding temperature measured
by the radiometer was recorded. The calibration covered a temperature range from 25-

100 °C and the results are presented in Figure 5.5.

 

110

 

I I T I f
— Tgcorrected) - 1.00801(meaeured) - 0.307
100- -

80- -
70- ~
60- -

50- -

corrected temperature [ C]

ISO-I ..

 

20

 

f H I I T i I I
20 30 4D 50 60 70 80 90 1 00 1 l 0

measured temperature [ C]

 

 

 

 

Figure 5.5. Corrected temperature of thermocouple vs. measured
temperature of radiometer.

81

In Figure 5.5 the thermocouple temperature (assumed to be the true temperature)
is plotted as a function of the temperature measured by the radiometer. A linear curve

fit was performed on the data resulting in the equation,

Tcm=1 . 008*ng-0 . 307 5.1

where Tom is the corrected temperature measured by the thermocouple and Tm is the
temperature measured by the radiometer in °C. Although the constant term, ~O.307,
creates a difference between the actual temperatme and the measured temperature, it has
no effect on the relative temperature rise, which is the quantity needed to determine the
thermal conductivity of the diamond film. Since the relative temperature rise is quantified
as the difference between the initial temperature and the steady state temperature, the

constant drops out and we are left with

AT

cczA T 5.2

111968

Even though the exact value for the emissivity of the plate must be assumed to measure
the temperature of the plate with the radiometer, the internal calibrations used by the
control unit are assumed to be valid since the calibration is linear and has a slope of
1.008, which is less than 1% different from one. The recorded temperature range of 25-
80 °C extends beyond the expected relative temperature rise of the experiment.

The biggest challenge in using this type of temperature acquisition system,
however, involves the emissivity. In order to measure the absolute temperature or the
relative temperature rise of a material, the emissivity must be known or determined.

Unlike thermocouples and resistance thermometers, the temperature measured by the

82

radiometer is a function of the radiant intensity. For a simplified illustration, the emissive

power of a grey body is expressed as,

E=80T4 5.3

where e is the emissivity, o is the Stefan-Boltzmann constant and T is the temperature
in Kelvin (Siegel and Howell 1981). The radiometer internally calculates temperature
from the measured emissive power E. From equation 5.3 it is evident that the emissivity
must be used to determine the absolute temperature and we find that the same holds true
for the relative temperature rise. Using equation 5.3, the relative temperature rise of a

grey body is calculated to be

1 a a
80

cult-s

) 5.4

where T and E are the temperature and the radiosity at a certain state and T. and E. are
the temperature and radiosity, respectively, of the surroundings. From equation 5.4 we
also see that the relative temperature rise is a function of the emissivity.

In order to check the assumption that the relative temperature rise is a function of
the emissivity, the radiometer was used to record the temperature of a heater at two
different power levels. At each level, temperatures were measured using two different
assumed emissivities. Since the change in the radiosity (AB) is the same in each case,
the temperature difference at each emissivity is compared. Figure 5.6 is a plot of the
surface temperature of the heater for both high and low radiosity levels using emissivities

of 1.0 and 0.80.

83

 

 

100
flava) - 05.4
“ f~ r'\ l"
90" .1
85- _

Heater Temperature [ C]

 

 

 

 

Location on Surface of Heater

 

 

 

Figure 5.6. Heater surface temperature vs. location

At each level the average temperature across the surface of the heater was determined.
The small peaks correspond to the heater element locations across the surface. If the
temperature rise is a function of emissivity, we should get different AT’s at each
emissivity setting, which is the case. The temperature temperature difference at e = 0.80
(95.4-74.4) equals 21 °C whereas the temperature difference at e = 1.0 equals only (76.0-
58.6) 17.4 °C. These values of 21.0 and 17.4 are quite different, hence, it is evident that
in order to accurately determine the relative temperature rise across the surface of the
sample we must carefully determine the emissivity of the doped diamond film.

Procedures involving the determination of the emissivity are discussed later in Chapter

6.

84

5.3 Experimental Setup

A series of steady state experiments was conducted on one of the diamond film
samples (sample A) which was prepared by the Microstructues Laboratory. Because of
the special design considerations of the samples, the sample setup consists of only four
items:
I the diamond film sample,
I electrical conducting adhesive tape,
I electrical lead wires and

I an adjustable clamp to suspend the sample in a vertical position.

5.3.1 Preparation of the Sample Setup

Each of the two setups for each sample is constructed in exactly the same manner
starting with cutting the electrical conducting tape into two strips approximately 2 cm x
0.45 cm. It is important that the tape contact only the surface of the doped diamond film
and not the silicon substrate. Since good contact between the tape and the film surface
is imperative silver paint was added to the diamond-tape interface. A schematic and
photograph of sample A after experimental sample preparation are displayed in Figure 5.7

and Figure 5.8.

85

 

 

 

 

 

 

 

 

 

 

Figure 5.7. Schematic view of sample setup.

 

   
 
 
  
 

silicon
substrate

electrical
ey . - nnfiimflnn
Figure 5.8. Photograph of the sample setup.

 

 

Chapter 6

Procedures and Results for
One Dimensional Radial Experiments

This chapter presents the experimental procedures and preliminary results for the
measurement of the thermal conductivity for a doped diamond film sample prepared by
the Microstructures Laboratory at Michigan State University. As described in Chapter 4,
the one dimensional radial heat flow model was designed to decrease the relative error
in the measurements and to obtain a more symmetrical temperature distribution. By
choosing a diameter of less than 0.003 m for this model, convection, which becomes less
than 1% of the total heat generated, can be neglected, further simplifying experimental
procedures. Issues concerning the determination of diamond film emissivity, electrical
power, optimal radiometer settings and temperature acquisition are also discussed in this

chapter.

6.1 Determining the emissivity
As explained in Chapter 5, the emissivity of the diamond film must be determined

in order to accurately convert the emissive power measured by the radiometer to surface

86

87

temperature. The emissivity of the diamond film sample was determined by adjusting the
emissivity value input required by the radiometer such that the temperature calculated by
the radiometer matched that of a thermocouple attached to the surface of the diamond

film. The setup used to determine the emissivity of the diamond film is presented in

Figure 6.1.

    

Figure 6.1. Setup used to determine the emissivity
of the diamond film for sample A.

The area mode on the control unit of the radiometer was activated allowing for a real
time averaged temperature measurement over a small surface area anywhere within the
imaged field of view. With the small area positioned on the junction of the thermocouple

itself, the radiometer measured the temperature of the thermocouple as the sample was

88

heated over a temperature range from 25-55 °C. The temperatures matched when using
an emissivity of 0.63 as input into the control unit. The correspondence between the
temperature measured by the thermocouple and the temperature measured by the

radiometer using an emissivity of 0.63 is presented in Figure 6.2.

 

 

2 I t-temperoture meaeured by Wale 2
55‘. — r - 1.00M - 0.277 1
I — e-O.” ‘

Temperetweolmdiometedcl
‘7’ a

114 IAAJ‘JIIAL AIAA

N
U

 

 

 

 

N
O

 

Temperature of thermocouple [ C]

 

 

Figure 6.2. The relationship between the temperature measmed by
the radiometer and the thermocouple at an emissive setting of 0.63.

By performing a linear curve fit on the data, we see that there is a slope of 1.007
for a temperature range between 25-55 °C, refer to Figure 6.2. The correlation coefficient
for this linear curve fit was determined to be 0.998. Since there is less than 1% error in
assuming a slope of 1.000 and since the constant offset is cancelled out when taking
differences, we are confident in assuming the emissivity of the diamond film to be 0.63

over the temperature range of interest here.

89

Ideally, we would like to determine the emissivity with the junction of the
thermocouple attached to the free-standing diamond film, since it is the temperature
distribution in this region which will be recorded. However, attempting to apply and
attach a thermocouple to this fragile area proved to be disastrous for sample B. Upon
first attempting this procedure using sample B, the free-standing diamond window was
subsequently broken when thermocouple wires (approximately 100 pm in diameter) were
pushed through this region. With just one other sample to work with, sample A, every
precaution was taken to ensure its structural integrity not only in this procedure but in the

ones that followed.

6.2 Measurement of the Electrical Power

Measuring the electrical power in the designed circuit is imperative in determining
the volumetric heat generated in the doped diamond film region. Without knowing the
volumetric heat generation it is impossible in the present experiment for us to determine
the thermal conductivity of the CVD doped diamond film. However, due to the rapid
thermal response of diamond during heating, the Omega digital multimeter can have
difficulty tracking the electrical current flowing through the circuit. This problem is
further compounded by the fact that electrical power generated within the doped region
of the film was found to be a function of temperature. For this reason, the electrical
power is calculated by determining the electrical resistance of the doped film as a
function of temperature, rather than the standard method of measuring the voltage across
a known resistance. In order to measure the resistance as a function of temperature the

sample was placed in an oven where the electrical resistance was measured using the

90

digital multimeter and the temperature was measured by the thermocouple used to
determine the emissivity. The temperature of the oven was varied over a range from 25-
55 °C. The results of the measured resistance at different temperatures are presented in

Table 6.1.

 

 

 

 

 

 

 

 

Temperature (°C) Resistance ((2)
25 1184
30 1 173
35 1 157 i
40 1 144
45 1 124
50 l 101
55 1075 i

Table 6.1. Temperature versus resistance for sample A.

Notice that the resistance decreases with increasing temperature, which is characteristic
of semiconductors. About a 10% reduction in the electrical resistance is noted over a

temperature rise of 30 °C.

91

Hence, instead of measuring the current through a known resistor and calculating
the power using V*I, where V is the voltage and I is the current, the electrical power can
be more accurately determined in this case from the V2/R relationship, where V is the

voltage and R is the electrical resistance.

6.3 Radiometer Settings

After the emissivity (e) and the resistance of the diamond film have been
detemrined, the radiometer is now prepared for thermal image acquisition. Using an
emissivity of 0.63, the thermal sensitivity was maximized by using the smallest
temperature range of 5 °C. Four images were averaged in real time to improve the signal-
to-noise ratio. Image averaging is an attractive option for two main reasons. First and
foremost, the signal-to—noise ratio alone is increased by a factor of two in this case and
second the amount of data is compressed by a factor of four. However, when using this
real time image averager, one must be certain that the displayed thermal field, a
composite image averaged from fom separate fields, does not contain any thermal
information of the relative temperature rise of the diamond window in its transient state.
Since the averaged image is being scanned at a 0.07 sec interval (60/4 Hz), we need to
capture the image at least 0.07 sec after the relative temperature rise is at a steady state
condition or transient effects would be averaged into the results.

The time constant of the film can be approximated by the relation,

tel”: , 6.1

a

(Carslaw and Jeager 1959) where at is the thermal diffusivity of the diamond film, b is

92

the radius of the etched region and t is the approximate time in which the relative
temperature rise reaches steady state. Assuming a thermal conductivity of 200 W/m K,
a density (p) of 3500 kg/m3 and a specific heat (CI) of 509 J/kg K, the thermal diffusivity
(0t) is calculated to be 112.3 x 10 ‘ m2/sec for this case where b = 0.003 m. A thermal
conductivity of 200 W/m K, which is on the low end of the published values for the
thermal conductivity of diamond films, was used to give us a "worst case" scenario for
the time constant. The larger the time constant, the more limiting is the use of the image
averager. From equation 6.1, the time constant of the diamond film is calculated to be
approximately 0.02 sec using a radius (b) of 1.5 mm. Hence, as long as we capture the
image 0.10 see after the film has started to heat, the composite thermal field averaged
over four images should represent the relative temperature rise in its steady state
condition.

It should be noted that although the relative temperature rise of the etched
diamond window reaches a steady state condition, the surface temperatures continue to
rise as the sample is heated. This condition is termed a quasi-steady state and it is this
condition that describes the global heating of the diamond-silicon specimen.

Because the heating of the sample is described by a quasi-steady state condition
and not entirely by the steady state condition modeled, it is important to determine the
amount of energy lost due to the heat capacity of the diamond sample. The effective
volumetric heat capacity gm, of the diamond film which is the energy absorbed in the

film itself is given by

BT
ghcap= P C335; 6-2

93
Using the density (p) and the specific heat (CI) for diamond, as previously determined,

and typical experimental data where aT/at = (30-22)°C/ 2 secs, gm, ~ 891le W/m’.
We now need to compare the volumetric heat capacity, gm with the measured volumetric
heat generation, gd. For an experiment corresponding to the particular temperature rise
used to calcuate gm, , g,l as 1.75x109 W/m’. By determining the ratio (ng/gd), we see
that the energy needed to increase the temperature of the exposed diamond film in its
quasi-steady state represents less than 0.1% of the total volumetric heat generation (g)

in the film.

6.4 Surface Temperature Measurements
Acquiring surface temperature measurements when using the thermal image
processor consists of two main processes: 1) capturing the desired thermal field and 2)

acquiring the surface temperatures, respectively.

6.4.1 Capturing Images Using the Thermal Image Processor

Before the actual thermal images are acquired, the sample must be aligned and
focused within the desired field of view (FOV). Because the focal distance between the
sample and the scanner is approximately 15 cm, the minimum FOV could not be used
because the scanner aperture reflected back off of the sample. This reflection shows up
as a black dot, which is present in the center of the image at close range and is magnified
when decreasing the FOV. As a result, the FOV and the position of the sample were
adjusted to avoid this reflection. In the experiments that follow two different FOV’s were

utilized.

94

Once the optics are adjusted and the scanner focused, the sample was connected
to the electrical circuit and a constant voltage is applied. The temperature of the sample
immediately increases and as it enters into the user defined range (e.g. 25-30 °C) the
image is captured and stored using the freeze frame on the image processor. It is this
image which contains the desired temperature distribution across the film.

During this procedure, the radiometer was first focused on the front side of the
sample. However, in this position the boundary temperature at the silicon-diamond
interface was difficult to determine. Since the chemical etches the silicon at an angle, the
thickness of the silicon at this interface is relatively thin, making it difficult to locate
when imaging through the diamond on the front side. A photograph of the sample setup

viewed from the front side is presented in Figure 6.3.

    
 
 
  
    
 
   

silver paint

diamond
' w1ndow

electrical
conducting
tape

 

Figure 6.3. Sample setup viewed from the front side.

95

As seen in Figure 6.3, silver paint was used on the sample to increase electrical contact
between the electrical conducting tape and the doped diamond surface, resulting in
increased uniform heating. The paint was not essential.

In order to locate the boundary of the free-standing diamond window 3 mm in
diameter with greater precision, the sample was simply rotated 180° such that the back
side of the sample was imaged. A photograph of the sample setup in this position is

presented in Figure 6.4.

etched
diamond
window

 

 

 

Figure 6.4. Sample setup viewed from the back side.

96

In this position the diamond-silicon interface can be determined more accurately in the
infrared image. Since there is a negligible temperature gradient across the thickness of
the diamond film and since the sample is symmetric on both faces, the temperature
distribution on both sides of the sample is assumed to be equivalent. Hence, our results
should not be adversely affected by positioning the sample in this manner.

Before acquiring the thermal images, the precise location of the diamond- silicon
interface in the FOV is determined. By using the electronic ruler provided by the image
processor software, the coordinates for the diameter of the free standing diamond film are
located. After the diameter is located, the x,y coordinates which are displayed by the
RGB monitor are recorded such that the appropriate temperature distribution can later be
extracted from the acquired thermal field.

After the diameter of the free standing film has been located in the FOV, the
thermal fields can now be acquired. With the power settings set at approximately 0.40,
0.62 and 0.88 watts five thermal images were captured using two different FOVs. These

images can be viewed in Figures 6.5-6.9.

97

,(‘mez-ei 23911 f. a

       
  
    

  

   

4+ "i; ‘53..»35 (13:35 :24; a.vg=-.;-¢, 2'3 NEE-E Fez:
Figure 6.5. Image 1 from Figure 6.6. Image 2 from
experiment #1 at 0.40 watts. experiment #2 at 0.62 watts.

'2‘? E“: 1’51».th ”SEE ElfITT'J’aiE— £7 *3} 157‘):
Figure 6.7. Image 3 from
experiment #3 at 0.88 watts.

 
   

24 out Erl-‘I‘JE 2x35: Emmi-:5: r53 .39 etc as 57-: raw-.55 m3; 3.... .. .
Figure 6.8. Image 4 from Figure 6.9. Image 5 from
experiment #4 at 0.40 watts. experiment #5 at 0.88 watts.

98

6.4.2 Temperature Acquisition

After the thermal images have been acquired using the appropriate control unit
settings on the radiometer, the temperatures can be determined from the thermal field
using the File_Access program run by the GWBASIC interpreter. This program, which
is part of the thermal image processor software, extracts data from a particular line in the
thermal field and converts the individual pixel intensity to temperature using calibration
tables stored in the ROM of the radiometer’s microprocessor. To extract the temperature
distribution, File_Access simply prompts the user for the image filename and the line
number. With the location of the diameter already determined, the line number is entered
and the temperatures corresponding to the pixel location across the entire thermal field
are then written to a data file.

After the temperatures have been determined across the designated line from the
thermal field, the next step is to extract only those temperatures associated with the
temperature distribution across the diameter of the diamond window. Since the
coordinates of the diameter were already determined by using the electronic ruler, the
temperature distribution across the 3 mm diameter free-standing diamond window for
each of the five experiments can be located and extracted from the line temperature data
extending across the entire horizontal FOV. The temperature distribution for each of the

five experiments can be seen in Appendix B.

6.5 Measurement of the Thermal Conductivity
Quantifying the thermal conductivity after acquiring the temperature distribution

across the diamond film requires two steps. The first step is to develop a mathematical

99

model for the problem; and the second step is to use the temperature data and the model

together to estimate the thermal conductivity (k, I)°

6.5.1 The Simplified Experimental Model

Because the temperature difference between the smface of the film and the
surroundings is small (within 10 °C), it is reasonable to assume that radiation can be
neglected. The validity of this assumption is demonstrated below through acquired

experimental data. Heat transferred by radiation is described by

grad=2Aeo (T‘-T,,‘) 6.3

where e is the emissivity, 0‘ is the Stefan-Boltzmann constant, T is the temperature of the
film, T. is the temperature of the surroundings and 2A (21:11) represents both sides of the
surface area of the exposed diamond window. With 8 = 0.63, 0' = 5.67 x 10" W/m2 K‘,
r = 0.0015 m, T = 305 K (32 °C) and T, = 295 K (22 °C), qml = 0.00054 W. Since the
temperature difference used in this calculation simulates experiments for the largest heat
dissipation (q = 0.88 W) radiation represents less than 0.1% of the total heat generated
within the film and, therefore, can be neglected.

Since the diamond window was etched to only a diameter of 0.003 m to aid in the
strength of the sample, the effects of convection, discussed in detail in Chapter 4, can also
be neglected With a characteristic length of 0.003 m convection S 1% of the total heat
generated within the film, when assuming a thermal conductivity between 250-1000 W/m

K, which is the expected range for the thermal conductivity of doped diamond. Tire

100

percentage of heat lost due to convection is, again, calculated from equation 4.19.
By neglecting convection, the former mathematical description of the experiment,
described by equation 4.10, is further simplified to

= gdbd 2_ 2
0(1) 4k(0u+6d) (b r) 6.4

 

where

V2
= 6.5
9" 12wa d

 

and

6(1) =T(I) -Tb 6.6

Equation 6.6 represents the difference between the temperature at the boundary

(r=b=0.0015 m) and the temperature at a particular radial position.

6.5.2 Estimating the Thermal Conductivity
Now that the appropriate temperature distributions have been acquired and the
governing equation simplified, the program NLIN (Beck 1993) is used to analyze the data

and estimate the thermal conductivity. NLIN uses the method of least squares

N
5:; [Yj-Tj12 6.7
=1

where N is the number of data points, Yj is the measured temperature and T1 is the

calculated temperature represented by

101
Tj=Tb+HK[1-(—Ig)3] 6.8

where H = (gd84b2)/4(5,,+84) and K = Me. Using the method of least squares the program
simultaneously estimates both T, and k to calculate Tj such that the error in S is

minimized. In order to minimize the error in S, NLIN uses the system of equations

as =_2" ”41:0 6.9
'33) g; r j
and
_a§=_ N - .. £1 2 = 6.10
6k 2; [Y1 leHtl (bH 0

Actually, NLIN is a sequential nonlinear estimation program that has more complex
calculations than indicated by equations 6.9 and 6.10 but these equations form the basis
of its estimation process for this linear problem.

By inputting H = V2b2/4RLW(5,,+84) for each case where L = 0.0045 m, W = 0.02
m, b = 0.0015 m and V is measured and R is determined from Table 6.1, Tj is calculated
for each of the five experiments. The measured temperature distribution (Y1) and the
corresponding calculated temperature distribution TJ across the surface of the diamond
window are displayed together in Figures 6.10 and 6.11. The largest temperature rise is

approximately 3 °C.

102

 

 

 

 

 

 

343 I I I I I I I
4

'5'
5
g a-OAO eette
S 25.0-

24.0- -

22.0 I 1 I I I I I

--l.00 -.75 -.50 ~25 0.00 0.25 0.50 0.75
r/b

 

1 .00

 

Figure 6.10. Measured and calculated temperature

distribution for experiments 1,2 and 3.

 

 

q-OAO eatte

Temperature [ C]

24.0-

34.0 I I I I I I I

i
32.0- ”-0... -'
30.0- -
28.0-

26.0- all
«i J

1

 

 

22" I I I I I
-1.00 -.75 -.50 -.25 0.00 0.25

r/b

 

I
0.50

I
0.75

 

1.30

 

Figure 6.11. Measured and calculated temperature

distribution for experiments 4 and 5.

 

 

103

NLIN not only fits the mathematical model, equation 6.4, to the data but it also
calculates the residuals which are the differences between the measmed and calculated
temperatures [Yj-Tj]. The residual distribution for each of the five experiments can be
seen in Figures 6.12-6.16. The standard deviations are approximately 0.18 °C, compared

to about a 2 °C temperature rise.

 

 

 

 

 

0.8 I l I T I I I
~ experiment It. q-O.4D watts r
0.6-e ..
0.4- --
'8 0.2-
.E
o 0.0-
s . 1
3. -.2- .
a 1
-.4-' -I
-.6- -
- J
’3 I I I I I I I
-1.00 -.75 -.50 -.25 0.00 0.25 0.50 0.75 1.00
r/b

 

 

 

Figure 6.12. Residual distribution for experiment #1, 107 pts.

104

 

 

 

 

 

0.0 I I I I I I I
- experiment '2. tar-0.62 watts '
0.0-I -
0.4- -
'5' 0.2-
H '1 q
‘3’ 00- -
g -.2- -r
1
-.4-1 -
1 I
-.6- -
“'8 I I I I I I I
-1.00 -.75 -.50 -.25 0.00 0.25 0.50 0.75 1. 30
r/b

 

 

 

Figure 6.13. Residual distribution for experiment #2, 107 pts.

 

 

 

 

 

0-3 I I I I I I m
< experiment '3. a-0.88 matte -
0.6- ..
0.4-1 -
4
'8 0.2 '1 _.
"" 1
g 00- -
:3
2° - ..
3 -2. .
-.4- d
‘ 1
-.6- -
‘3 I I I r I I I
- 1 .00 -.75 -.50 -.25 0.00 0.25 0.50 0.75 1 .00
r/b

 

 

 

Figure 6.14. Residual distribution for experiment #3, 107 pts.

105

 

 

 

 

 

0.8 I I I I I I I
- experiment [4. q-0.40 watts .
0.0- ..
0.4- -
q
8' 0.21 .
2:; . ‘
g 00- ..
i ‘ ‘
-1. ..
-.4‘ -
-.0-1 -I
”-8 I I I I I I I
-1.m -.75 -.50 -.25 0.“) 0.25 0.50 0.75 1.00
r/b

 

 

 

Figure 6.15. Residual distribution for experiment #4, 121 pts.

 

 

 

 

 

 

0.0 I I I I I I I
experiment '5. q-0.88 eatte -
0.0-1 ..
0.4- -I
'5' 0.2-1
-- A
0
'g 0.0- -
29 . .
é —.2~ -
-.4- e
--6-' -l
I J
'5 I I I I I I I
—1.00 -.75 -.50 -.25 0.00 0.25 0.50 0.75 1.130
r/b

 

 

 

Figure 6.16. Residual distribution for experiment #5, 121 pts.

106

Across most of the diameter, the residual pattern is unique for each of the five
experiments, representing a random rather than a systematic error. However, in the
region 0.75<r/b<1.00, a trend may exist. This trend could represent a systematic error
due to the non-uniform heating noticed while running the experiments. At certain
temperature ranges a slight thermal wave starting outside the region r = +b and continuing
across the diamond film was visible in the displayed thermal field. This wave could be
the result of non-uniform doping in the semiconducting diamond layer, non uniform
thickness in the diamond or even the non-uniform thickness of the silicon at the boundary
where r = b.

Although the pattern is mostly random from one experiment to another, the
residuals remain correlated due to the number of points making up the "peaks" and
"valleys" describing the residual distribution.

The thermal conductivity is estimated sequentially from the temperature data
across the surface of the entire diamond window using the system of equations presented

by equations 6.8, 6.9 and 6.10. The results are presented in Figures 6.17-6.21.

107

 

 

450-] experiment [1. tau-0.40 watts

I‘ll

lllLLl
l

\lllllllljlllAll l

i

Thermal Conductivity [VI/m K]
N
8
l

 

‘ Alllllljllllllllllljlll

 

e

_e
o-
_e

0"1fl'l'l'1'lf

11 21 :51 41 51 61 7'1 ' 8'1 ' 9'1
number labeled data across diameter

 

 

 

 

Figure 6.17. Sequential estimation of the thermal conductivity
for experiment #1, 107 data points.

 

 

 

 

 

 

5m I j I ' I l I ' I ' 1 ' I ' I ' I :
450 experiment [2. a-O.82 watts 1
5 3
350 -‘
E. a
3. 300 i
I; s
g 250 1
‘D M
g 200 -j
.5 I
150 1
E :
IS 100 r:
50 “I

0 I ' I ' I ' I ' I ' I m I I I ' I I ' r

1 11 21 31 41 51 51 71 81 91 101

number labeled data across diameter

 

 

 

Figure 6.18. Sequential estimation of the thermal conductivity
for experiment #2, 107 data points.

108

 

 

V I V r 'i I j ' V I U I V I I I V ' V l V

W '3. 4.1-0.03 eatte

 

AAllAALAlAAA‘lAAAAIAIAAIA‘AA

All

I‘Al

ThemalConductmtyDl/mlt]
.ee‘giiseis

 

 

A “ALIAAIAIA

l I I l I l l T
11 21 31 41 51 61 71 81 91 101
number labeled data across diameter

 

 

 

Figure 6.19. Sequential estimation of the thermal conductivity
for experiment #3, 107 data points.

 

 

 

 

 

 

m : I I I I I I I I ‘U l V l I I I l I I I I V I I :
450.: W‘ '4, a-0.40 watt! '3
€ 350% -2
3.. 3 E
:5 3001 1
'75 250.: .1
5 2W1 1
E 150-: i
1 1
IE too-j -§
50-2 -I
0 ‘ ' r I I ' I ' I ' I ' I ' I ' I ' I ' I I I ' ‘

‘ ll 21 31 41 51 61 71 81 91 101 111 121

number labeled data across diameter

 

 

 

Figure 6.20. Sequential estimation of the thermal conductivity
for experiment #4, 121 data points.

109

 

 

‘8'

rI'l'rjl'l'l'l'l'l‘l

All]

A
2?

experiment I5. q-0.88 watt:

.5
5.3

All All]
llllJlllAllJ

u
*5

All

(A
E?

 

1.11 llll l

lllllljll

Thermal Conductivity [VI/m K]

_. N
8 8
1 1

1 1 1... 1 11 11.4 1

08

_A lilllllll

 

I I I [TI I I I l I ' I l I I I I I I I I I.
11 21 31 41 51 61 71 81 91 1011111221

number labeled data across diameter

 

 

 

 

 

Figure 6.21. Sequential estimation of the thermal conductivity for
experiment #5, 121 data points.

The initial estimates for the thermal conductivity using the first 25 points or so
vary with great magnitude. This variation is due to the limited amount of data for the
system to converge to a consistent estimate for both Tb and k. As more data are used,
however, the estimate begins to settle around a tighter range and remains relatively
constant throughout most of the second half of the data set.

The final estimate either at x=107 or x=121, depending on the particular FOV used
in the experiment, represents the best estimate by the system for the thermal conductivity.
The estimated thermal conductivity for each of the five experiments is presented in Table

6.2

110

 

 

 

 

 

Experiment k 1 Voltage Resistance Power
No. (W/m K) (volts) (ohms) (Watts)

1 242 21.8 1 184 0.40

2 225 27.1 1 177 0.62

3 256 32.1 1173 0.88

4 236 21.7 1184 0.40

5 241 32.1 1173 0.88

 

 

 

 

Table 6.2. The thermal conductivity estimated by NLIN for each of the five
experiments at different power levels.

For the five experiments the average value for the thermal conductivity for sample
A and the standard deviation from that average are calculated to be 240 :t 11 W/m K,
respectively. The standard deviation represents experimental reproducibility to within
a 5% variation in the average value for one sample using three different power levels and
two different fields of view (FOV).

The thermal conductivity for sample A is determined from experiments 3 and 5
where the electrical power was set at 0.88 watts and the signal-to-noise ratio is greatest.

The 100(1-a)% confidence interval for the resulting estimates in each experiment is

111
assumedtobe

easy—(FEW 1:11,, (n-p) sksk+as (x’x) ’ c,.,,,2 (n-p) 6.11
where o, is the standard deviation of the temperature, \/((XTX)") is the square root of the
covariance matrix divided by the variance and tum (n-p) is the t statistic for (n-p) degrees
of freedom (Beck and Arnold, 1976). For 95% confidence tm5(100)zl.998 (Miller and
Freund, 1965). Since a, and «((xTxyl) are calculated from the NLIN output file, the
estimated thermal conductivity and confidence interval for experiment #3 and experiment
#5 are determined to be 256 i 9 W/m K and 241 :l: 7 W/m K, respectively. The
uncertainty of the measurement is approximately (lOOx(9/256)%) 3.6% for experiment
#3 and (100x(7/24l)%) 3.0% for experiment #5.

Hence, the estimated thermal conductivity for sample A, calculated by averaging
the results obained in experiments 3 and 5, is determined to be 249 :1: 8 W/m K. The
confidence interval represents a 3.2% uncertainty in the estimated value of the thermal

conductivity.

6.6 Experimental Uncertainty

The experimental uncertainty is a measure of the accuracy of the method used in
determining the thermal conductivity and accounts for measurement uncertainties in
quatities such as thickness of film, voltage, temperature, etc. It acts as a global
uncertainty which is driven by the precision at which all of the experimental unknowns
can be measured. The uncertainty in the measurement of the thermal conductivity is

determined from the expression,

112

6k 2+ 6k 2+ 2
wk= [(WAV) +(::Ab) +(—3::AR)
2 2 6.12

kA 2 + + 2 2
where the terms ak/8§ are the sensitivity coefficients determined from equation 6.4 and
the AE, terms are the estimated errors in the measurements. The experimental uncertainty
is calculated using the power levels and temperature rises for experiments 3 and 5, as
stated before. The experimental parameters, their uncertainties and the corresponding

contributions to the uncertainty of the thermal conductivity are presented in Table 6.3.

 

 

 

 

 

 

 

 

 

I Experimental Parameter Estimated Uncertainty Contribution to the uncertainty
(é) in Parameter mg) in the thermal conductivity
(Wz/rn2 K2)

V = 32.1 volts AV = 0.1 volts (alt/3V AV)2 = 3.0

R= 11739 AR=SQ (Bk/3R AR)2= 1.4
b = 0.0015 m Ab = 0.00001 m (alt/ab Ab)” = 13.7 ,
l L = 0.0045 111 AL = 0.00010 m (ak/aL AL)2 = 38.0 I
w = 0.02 m AW = 0.00005 m (art/aw AW)2 = 0.5 '

l 0 = 3 °c A9 = 0.1 °c (ak/ae A0)2= 85.4
5,+6, = 0.0000066 m A(5,+8,) = 0.0000001 m (ak/a(5,+5,) M5, 2 = 17.65

Table 6.3. Experimental parameters, their estimated uncertainties and the
corresponding contributions to the uncertainty of the thermal conductivity.

113

The uncertainty in the voltage measurement (AV = 0.1 volts) was chosen since the
DVM recorded voltages to the nearest 0.1 volts. The uncertainty in the resistance (AR
= 5 Q) was determined from the variation of the resistance due to temperature range from
28-31 °C. The uncertainty in the radius (0.00001 m) was determined by measuring the
sample under a microscope using an objective micrometer which had a sensitivity of
0.00001 m. The uncertainty of the width :1: 0.00005 m and the uncertainty in the length
of 0.00010 m is due to the silver paint which was added to the surface of the sample to
improve the electrical contact. The boundary of the silver paint is not as sharp and clear
as the other recorded dimensions. The uncertainty of the diamond film thickness was
determined from the sensitivity of the SEM and a conservative estimate for the
uncertainty of the relative temperature rise, 0.1 °C, was also chosen. By substituting the
values presented in Table 6.3 and evaluating equation 6.12 the thermal conductivity along
with the experimental uncertainty is calculated to be 249 :l: 13 W/m K. This represents
approximately a 5.2% uncertainty in the estimated value of the thermal conductivity. It
is interesting to note that the confidence interval calculated by NLIN, which represents
uncertainties in the relative temperature distribution, and the experimental uncertainty
calculated from only the uncertainty in temperature are nearly equivalent assuming 0 =
3 °C and A0 = 0.1 °C. The confidence interval was calculated to be i 8 W/m K while
the experimental uncertainty due to only the temperature was calculated to be :1: 9 W/m
K. We see from Table 6.3 that although the experimental uncertainty is primarily driven
by the uncertainty in temperature it is also affected by uncertainties in other experimental
parameters such as length, radius, etc.

After examining the previous works concerning the thermal properties of diamond

114
films discussed in Chapter 2, there are only two studies which determined the

experimental uncertainty. Graebner, Jin, Kammlott, Bacon, Seibles, Banholzer (1992) and
Graebner, J in, Kammlott, Herb, Gardinier (1992) determined the experimental uncertainty
for their laser pulse method to be ~3%. This uncertainty was reportedly determined by
the uncertainty in film thickness. One advantage in using the laser pulse technique is that
the heat dissipated does not have to be quantified and, hence, there is no need to take into
account heating power uncertainties. However, there are uncertainties in the temperature
measurements and in the pCP product, which enables the calculation of the thermal
conductivity from the thermal diffusivity. It is unclear in both articles if uncertainties in
temperature and in pCp are included in the results or if their reported uncertainty of 3%

is calculated only from the uncertainty of the diamond film thickness.

Chapter 7

Summary and Conclusions

The goal of this research was to design experimental methods and techniques for
the measurement of doped CVD diamond film thermal conductivity. In the pursuit of this
goal both analytical and experimental studies were conducted.

From the analytical investigation, the optimal experimental model was first chosen.
A sensitivity analysis determined from one dimensional, two dimensional and radial
potential geometries revealed that the radial heat flow model was most sensitive to the
measurement of the thermal conductivity. Further investigation of the radial model
revealed optimal geometrical parameters such as the diameter and thickness of the doped
diamond layer which would limit convective losses and decrease the relative error.
Parameters such as film resistivity, film geomeu'y and thickness of the undoped diamond
layer were also chosen to obtain the desired temperature rise.

As a result of the analytical analysis, two diamond film samples were fabricated
by the Microstructures Laboratory at the Michigan State University Research Complex
using hot-filament CVD deposition. A new experimental setup was constructed and its

design was the result of specialized sample fabrication techniques. A bi-regional

115

1 16
deposition process was conducted to deposit a doped (semiconducting) diamond layer on

top of an undoped (electrically insulating) diamond region. The undoped layer with a
thickness of approximately 1 pm ensured electrical insulation between the two
semiconducting layers (doped diamond and silicon). The doped layer with a thickness
of approximately 5.6 pm was fabricated by injecting boron into the film as a charge
carrier which provided a means of electrically heating the samples. A diamond patterning
technique was also used to deposit a diamond mask on the backside of both samples.
After the mask was deposited, the exposed silicon was chemically removed, creating a
free-standing diamond window. Because of the mask’s chosen geometry and the chemical
application process utilized during etching, the silicon was removed in the circular
configuration needed to simulate the optimal radial experiment.

Temperature measurements were recorded using an infrared thermography
technique. The temperature acquisition system, consisting of a high speed IR scanner,
control unit, RGB monitor and high resolution thermal image processor was utilized to
capture thermal images of the diamond sample during its quasi-steady heating.
Temperatures across an entire line spanning the horizontal FOV and corresponding to the
location of the diameter of the diamond window were first calculated from individual
pixel intensity using internal calibration tables stored in the ROM of the infrared control
unit and then extracted from each of the thermal images. The temperature distribution
across the 3 mm diamond window was located in the extracted temperature file using an
electronic ruler provided by the image processor software.

Five different experiments were conducted on one of the prepared samples using

two different fields of view resulting in 107 and 121 individual surface temperature

117
measurements at three different power levels (0.40, 0.62 and 0.88 watts). The thermal

conductivity was estimated in each experiment using the method of least squares to
minimize the error between the measured temperature distribution recorded by the infrared
temperature acquisition system and the calculated temperature distribution determined
from the optimal radial heat flow model.

For the five experiments the average value for the thermal conductivity for sample
A and the standard deviation from that average was calculated to be 240 i 11 W/m K,
respectively. The standard deviation represents an experimental reproducibility for a
single sample at three different power levels and two different fields of view to within
a 5% variation in the average value.

The thermal conductivity for sample A was determined from experiments 3 and
5 where the electrical power was 0.88 watts and the signal-to-noise ratio was greatest.
The estimated thermal conductivity for sample A, calculated by averaging the results
obained in experiments 3 and 5, was determined to be 249 j: 8 W/m K using a least
squares approximation. The confidence interval (:1: 8 W/m K) represents a 3.2%
uncertainty in this estimated value, but only considers uncertainties in temperature.

The experimental uncertainty which was determined by estimated uncertainties in
the measurements of all the experimental parameters is calculated to be i 13 W/m K.
Hence, the estimated value for the thermal conductivity of a doped diamond film using
the experimental methods and techniques developed in this work is determined to be 249
d: 13 W/m K, representing an experimental uncertainty of ~ 5%.

Based on the author’s literature review, this is the first reported value for the

thermal conductivity of a semiconducting diamond film.

118

Suggestions to r Future Work:

1) Determine the accuracy of the methods and techniques used in the present work by
simulating a similar sample setup using a material of known thermal conductivity (silicon)
and then compare the known and estimated values. This "dummy" sample setup can be
used to validate the mathematical model in different proposed experimental scenarios.

2) Make the sample more robust by depositing more diamond on the silicon substrate.
By increasing the thickness of the doped diamond film, a more reliable sample can be
fabricated, allowing for increased diamond window diameters, larger temperature rises,
increased spatial resolution, decreased convective effects and better estimates of the
thermal conductivity as determined by the sensitivity analysis. A thickness of at least 10
pm for the doped layer of diamond is suggested.

3) Measure the thermal conductivity as a function of the film thickness.

4) Measure the thermal conductivity as a function of the film resistivity (doping level).

5) Measure the thermal conductivity at high temperatures using a) the present setup
which is placed in an open environment which could be adversely affected by convection,
b) the setup placed in a vacuum which could be adversely affected by radiation and c)
the setup placed in an oven which may be most immune to the effects of convection and
radiation. The "dummy" sample setup can be used here to both determine the optimal
experimental environment for the measurement of the thermal properties at high

temperatures and to validate the mathematical model.

6) Use the transient capabilities of the IR temperature acquisition system described in
Appendix 1 to determine the density-specific heat product, pCp, for CVD doped diamond

films.

119

7) Develop a new experimental method which will not require specialized sample design
and extensive sample preparation for the measurement of the thermal properties (e. g. bi-
regional diamond deposition, diamond patterning, masking and etching). These
techniques are very time consuming and their permanent effects on the sample limit the
practical use of the diamond film after it has been prepared. Developing a method which
would use a laser line heat source instead of electrical heating would allow for the
determination of the thermal properties of both doped and undoped CVD diamond films

with minimal sample preparation.

8) Compare the results obtained by the method used in the present work with the method
proposed in 7.

9) Determine the thermal properties for diamond film samples prepared by the
microwave method and compare with the experimental results obtained from the diamond
film samples prepared by the hot filament method.

10) Determine the degree of reproducibility obtained for the thermal properties of
diamond films prepared by a fixed processing method.

List of References

Albin, S., Winfree, WP, and Crews, B. Scott, "Thermal Diffusivity of Diamond Films
Using a Laser Pulse Technique", J. Electrochem. Soc., Vol. 137, 1990, No. 6.

Anthony, T.R., Fleischer, J.L., Olson, J .R., and Cahill, D.G., "The Thermal Conductivity
of Isotopically Enriched Polycrystalline Diamond Films", J. Appl. Phys, Vol. 69,
1991, No. 12.

Baba, K., Aikawa, Y., and Shohata, N., "Thermal Conductivity of Diamond Films",
J. Appl. ths., Vol. 69, 1991, No. 10.

Beck, J.V., and Arnold, K.J., Parameter Estimation in Engineering and Science, John
Wiley and Sons, 1977.

Beck, J.V., Cole, K., Haji-Sheikh, Litkouhi, B., Heat Conduction Using Green’s
Functions Hemisphere Press, N.Y., 1992.

 

Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, 2nd ed., Oxford University
Press, London, 1959.

Graebner, J.E., Jin, S., Kammlott, G.W., Bacon, B.,Seib1es, L., and Banholzer, W.,

"Anisotropic Thermal Conductivity in Chemical Vapor Deposited Diamond",
J. Appl. Phys, Vol. 71, 1992, No. 11.

120

121

Graebner, J.E., Jin, S., Kammlott, G.W., Herb, J.A., and Gardinier, C.F., "Unusually High
Thermal Conductivity in Diamond Films", Appl. Phys. Lett., Vol. 60, 1992, No.
13.

Graebner, J.E., Jin, S., Kammlott, G.W., Herb, J.A., and Gardinier, C.F., "Large
Anisotropic Thermal Conductivity in Synthetic Diamond Films. ", Letters to Nature,
Vol. 359, 1992, p. 401.

Graebner, J.E., Mucha, J .A., Seibles, L., and Kammlott, G.W., "The Thermal Conductivity
of Chemical-Vapor-Deposited Diamond Films on Silicon.", J. Appl. Phys., Vol.
71, 1992, No. 7.

Holman, J.P., Exmrimental Methods for Engineers, McGraw-Hill, Inc., 1978.

Incropera, ER, and DeWitt, D.P., Introduction to Heat Transfer 2nd ed., John Wiley and
Sons, 1990.

 

Machlab, H., McGahan, W.A., and Woollam, J.A., "Thermal Diffusivity Measurements
by Photothermal Laser Beam Deflection (PTD): Data Analysis Using the
Levenberg-Marquardt Algorithm. ", Thin Solid Films Vol. 215, 1991, pp. 103-107.

 

Miller, I., and Freund, J.E., Probabilig and Statistics for Engineers, Prentice-Hall, Inc.,
1965.

Morelli, D.T., Beetz, OR, and Perry, T.A., "Thermal Conductivity of Synthetic Diamond
Films", J. Appl. Phys., Vol. 64, 1988, No. 6.

122

Ono, A., Baba, T., Funarnoto, H., and Nishikawa, A., "Thermal Conductivity of Diamond
Films Synthesized by Microwave Plasma CVD.", J_appnese J. ADDl. Phys., Vol. 25,
1986. No. 10.

Parker, W.J., Jenkins, R.J., Butler, CR, and Abbott, G.L., J. Appl. Phys., Vol. 32, 1679,
1961.

Siege], R., and Howell, J.R., Thermal Radiation Heat Transfer, 2nd ed., Hemisphere
Publishing Corp., 1981.

Spehar, J ., "Diamonds, Electronic’s Best Friend.", IEEE Potentials Vol. 10, No. 4, 1991,
pp. 9-12.

 

Vorobei, V.V., "Thermoelastic Problems for Monodirectional Fiber Composites Exposed
to Pulsed Thermal Action.", Soviet Applied Mechanics, Vol. 22, 1986.

123

Appendix A.

Calibration of the Inframetrics 600L Radiometer for

Rapid Transient Thermal Events

By connecting the signal output from the conuel unit to an AID board, the
radiometer, consisting of the high speed IR scanner and control unit, has the ability to
track transient thermal events. However, the voltage signal output from the control unit
which is proportional to temperature must be calibrated. The calibration of the 600L
radiometer for imaging transient thermal events as detailed in this section consists of five
components: 1) setup and execution, 2) reproducibility, 3) temporal resolution, 4) thermal
resolution and 5) relative accuracy. These components are discussed in turn in the

following paragraphs.

Setup and Execution:
Displayed in Figure A.l is a diagram of the experimental setup used in the

calibration.

124

 

 

 

 

control
unit

 

 

 

 

 

 

 

”JUNO!“

 

 

 

 

AT-MIO-16F5

 

 

 

 

 

 

 

 

 

 

Figure A.l. Setup used for the calibration of the IR radiometer to record
transient thermal events.

The setup consists of four main instruments: 1) The Inframetrics 600L IR radiometer,
2) black aluminum disk with heater and thermocouple, 3) computer data acquisition
system with the National Instruments (AT-MIO-16FS) AID board and 4) the Omega Type
E thermocouple thermometer, model 450-ABT.

Before executing the calibration, three settings have to be made on the radiometer.
The first setting is related to the emissivity of the small region surrounding the
thermocouple on the surface of the disc. The emissivity of the disc is selected by
matching the temperature measured by the radiometer with the temperature measured by
the thermocouple which is attached to the surface of the aluminum disc. As a
precautionary measure, this procedure should not commence until the liquid nitrogen

temperature and the dewar temperature have stabilized within the IR scanner or results

125

could be erroneous. This time is approximately 25 min.

After the thermometer and radiometer temperatures have been initialized by
selecting the appropriate emissivity, the fast line scan mode is selected from the control
unit. Since only horizontal scans occur in this mode, the thermocouple can be scanned
continuously at high frequencies by the radiometer. The calibration must be executed in
the fast line scan mode due to the vertical and horizontal line scans performed in the
regular imaging mode which make it extremely difficult to later recover the appropriate
voltage data values corresponding to the location of the thermocouple since the entire
FOV is being scanned. In order to ensure that the exact location of the thermocouple on
the surface of the disc was being scanned, the thermocouple was placed in a crevice
extending across the diameter of the disc and positioned vertically in the FOV. The
thermocouple was located as the darkest region in the FOV due to the higher emissive
value associated with the crevice. After the thermocouple was located a temperature span
of 50 °C was selected with a range from 15-65 °C.

The black aluminum heater unit was comprised of a type B thermocouple, an
aluminum disc and a 10 Q Minco thermofoil heater mounted on the backside of the plate
using a thin layer of silicon grease. The heater element was aligned parallel to the same
direction as the horizontal line scan to eliminate unwanted temperature gradients which
might distort the location of the thermocouple. The heater was connected to a power
supply where the temperature of the plate could be controlled.

Data was recorded on a CompuAdd 33 MHz, 486 PC which utilized the AT-MIO-
16F5 data acquisition board. The board which contains 12 bit resolution AID converters

was driven by standard LabWindows data acquisition software. The radiometer output,

126

a standard RS-170 signal, was connected to the computer by a coaxial cable where
voltage readings were recorded on the order of 1 volt. During execution of the calibration
the power to the heater was increased until the thermometer temperature reached a
maximum temperature of 65 °C. Every 2 °C, as the plate cooled, the data acquisition
program was activated and the corresponding voltages were immediately plotted. The
minimum voltage value in each of the scans which represents the temperature of the
thermocouple was averaged over several scans. The voltage measured by the computer
and the temperature measured by the thermocouple were then recorded and their
relationship determined.

A disadvantage of the calibration is that data must be retrieved manually making
the procedure quite time consuming, approximately 2 hrs/calibration. There are two main
reasons for the unusual time needed to perform this calibration: 1) since the radiometer’s
scanning process is continuous and not synchronized with the data acquisition, one cannot
predict at what time a particular line scan is recorded by the computer so the appropriate
voltage values remain inaccessible without a visual display of the acquired data and 2)
unwanted data values which come in the form of vertical refresh signals resemble

legitimate line scans and can best be omitted through visual examination, see Figure A.2.

Reproducibility:

This calibration procedure was performed six times over a five day period. A plot
of all six runs can be seen in Figure A.3. The standard deviation between the six
individual calibration runs and the average represents a reproducibility within 0.65 °C.

The emissivity which was determined for each individual calibration varied between 0.82

127
and 0.84.

Temporal Resolution:

The sample interval for the data acquisition system was set to its fastest rate of
227 kHz. The sample rate for the radiometer in the fast line scan mode is, however,
much greater at 8 kHz/line with a horizontal resolution of 256 pixels/line. This results
in a sample rate (pixels/sec) of approximately 2,048 kHz. The ratio between the
radiometer sample rate and the data acquisition sample rate of nearly 10:] results in a

resolution of 10 voltage values per line scan as seen in Figure A.4.

T hermal Resolution:

The thermal resolution represents the smallest recognizable temperature increment.
The smallest voltage increment detectable by the computer was found to be approximately
0.0024 volts. The calibrations represent an average sensitivity (slope) of 0.01 volts/°C,
see Figure A.2. From this we can calculate the system’s thermal resolution over a 50 °C

span to be 0.24 °C.

Relative Accuracy:

In light of the fact that the radiometer was calibrated using the Omega
thermometer as the temperature standard, we must base the success and usefulness of this
calibration on a relative scale where the temperature measured by the thermocouple is
treated as the accurate standard temperature. The relative accuracy which is defined here

by the standard deviation between the six runs and the quadratic curve fit describing the

128
average was calculated to be approximately 0.55 °C. The curve of best fit calculated by

PLOTIT was determined to be

T=—33 .65742*V2+130 .6025*V-2 .446104,

where T is the calculated temperature and V is the measured voltage. The relative
accuracy of :t 0.55 °C represents an error of approximately 1.4% (0.55/40) in this
particular calibration.

Once the temperature versus voltage relationship has been determined, it can be
entered into the PC data acquisition software allowing for voltage to temperature

conversion and ultimately transient temperature measurement capability.

129

Voltage (volts)

 

data count

Figure A.2. Vertical refresh signals resembling ligitimate line scans.

130

J
L)

 

(

0)
Ul

()1
C)

 

(l)
()1

(ll
C)

1. [J P P‘
U7 O (J1

lhtctmt‘xmuple lmnpcmlmt: l (1]

(cl
C.)

 

 

i\)
U1

J111141111141111111J111114411_1_1“11__,L_LJ-LL11111111L11

[\J
O
4

 

 

,CDJ ,1_1_1__1_14 1 1 111 111 1L1 11 1L1 1L1 1 L1_.l 111111_LJ_1.L.L1.J_LJ_L.1J_L.J.L_.

\J

" A, -e
J.‘ A
- \J u .1 v .4 U
.‘
WA!" A o’er \ ’K‘AA —\
4v v1 \.._.., .J by: (E:

Figure A.3. Calibration of the 600L IR Radiometer for rapid transient thermal events.

Voltage (volts)

 

data count

Figure A.4. Voltage output of radiometer during calibration.

Appendix B.

132

Temperature Distribution for Experiments 1-5

Experiment #1

1'

Tem .

-1.00000 25.33664

-.98113
-.96226
-.94340
-.92453

-.90566

-.88679
-.86792

-.84906

-.83019

-.81132

-.79245

-.77359
-.75472

-.73585
-.71698
-.6981 1
-.67925
-.66038
-.64151
-.62264

-.60377

-.58491
-.56604
-.54717

-.52830

-.50943
-.49057
-.47l70
-.45283

-.43396

-.41509
-.39623
-.37736
-.35849
-.33962
-.32076
-.30189

25.36899
25.27191
25.36899
25.36899
25.49594
25.52704
25.55817
25.58927
25.58927
25.65149
25.7 1372
25.71372
25.80707
25.83817
25.77594
25.96262
26.14929
26.21155
26.1 1820
26.02484
25.96262
25.99375
26.] 1820
26.24265
26.37207
26.33972
26.33972
26.37207
26.30734
26.27500
26.21155
26.05597
26.18042
26.307 34
26.40442
26.46915
26.563 14

-.28302
-.26415
-.24528
—.22642
-.20755
-.l8868
-.16981
-.15094
-.l3208
-.l 1321
-.09434
-.07547
-.05660
-.03774
-.01887

.01887
.03774
.05660
.07547
.09434
.1 1321
.13208
. 15094
. 16981
. 18868
.20755
.22642
.%528
.26415
.28302
.30189
.32076
.33962
.35849
.37736
.39623
.41509

Tem .

26.65674
26.75034
26.78156
26.78156
26.78156
26.68793
26.53192
26.43680
26.37207
26.62555
26.87515
27.06235
27.12479
27.12479
27.031 16
27.06235
26.90637
26.78156
26.81275
26.78156
26.71915
26.71915
26.75034
26.84396
26.87515
26.90637
26.87515
26.84396
26.81275
26.65674
26.46915
26.50073
26.59436
26.71915
26.71915
26.71915
26.78156
26.78156

r/b

.43396
.45283
.47170
.49057
.50943
.52830
.54717
.56604
.58491
.60377
.62264
.6415]
.66038
.67925
.6981]
.71698
.73585
.75472
.77359
.79245
.81132
.83019
.84906
.86792
.88679
.90566
.92453
.94340
.96226
.98113

Temp. (“Cl

26.68793
26.78156
26.71915
26.56314
26.53192
26.59436
26.68793
26.56314
26.5007 3
26.53192
26.43680
26.37207
26.21 155
26. 18042
26.1 1820
26.02484
25.96262
25.86929
25.83817
25.83817
25.77594
25.74484
25.65149
25.49594
25.40134
25.40134
25.40134
25.46481
25.55817
25.58927

1.00000 25.55817

133

Experiment #2
r Tern . rm Temp. (3) rip Temp. (3)

-1.00000 25.55258 -.07547 28.36267 .84906 25.97327
-.98113 25.90854 -.05660 28.39389 .86792 26.00559
-.96226 26.03797 -.03774 28.48749 .88679 26.03797
-.94340 26.13504 -.01887 28.51868 .90566 26.07031
-.92453 26.03797 .00000 28.51868 .92453 26.10266
-.90566 26.00559 .01887 28.51868 .94340 26.16739
-.88679 26.00559 .03774 28.48749 .96226 26.00559
-.86792 26.16739 .05660 28.48749 .98113 25.87619
-.84906 26.26074 .07547 28.6162] 1.00000 25.77908
-.83019 26.35407 .09434 28.61621

-.8ll32 26.44742 .1132] 28.51868

-.79245 26.59857 .13208 28.17548

-.77359 26.74860 .15094 27.92584

-.75472 26.83859 .1698] 27.67365

-.73585 26.89862 .18868 27.80106

-.71698 26.99109 .20755 27.98825

-.69811 26.99109 .22642 28.20667

-.67925 26.99109 .24528 28.30029

—.66038 26.89862 .26415 28.23789

-.64151 27.02362 .28302 28.08188

-.62264 26.95862 .30189 27.86347

-.60377 27.15360 .32076 27.80106

-.58491 27.31613 .33962 27.92584

-.56604 27.41364 .35849 28.05066

-.547l7 27.51114 .37736 28.11307

-.52830 27.60864 .39623 28.05066

-.50943 27.57614 .41509 27.95706

-.49057 27.44614 .43396 27.89465

-.47l70 27.54364 .45283 27.80106

-.45283 27.28363 .47170 27.73865

-.43396 27.28363 .49057 27.73865

-.41509 27.57614 .50943 27.64115

-.39623 27.67365 .52830 27.67365

-.37736 27.80106 .54717 27.51114

-.35849 27.89465 .56604 27.34863

-.33962 27.92584 .5849] 27.28363

-.32D76 27.98825 .60377 27.21860

-.30189 27.95706 .62264 27.25113

-.28302 27.92584 .6415] 27.28363

-.26415 27.76987 .66038 27.28363

-.24528 27.76987 .67925 27.15360

-.22642 27.89465 .6981] 27.12110

-.20755 28.05066 .71698 27.12110

-.18868 28.08188 .73585 27.05609

-.16981 28.05066 .75472 26.99109

-.15094 28.11307 .77359 26.89862

-.13208 28.20667 .79245 26.53858

-.11321 28.3“)29 .81132 26.26074

-.09434 28.26907 .83019 26.10266

Experiment #3

1'

-L00000
a98ll3
a96226
a94340
a92453
a90566
a88679
a86792
n84906
a83019
a81132
a79245
a77359
a75472
a73585
a71698
a69811
a67925
a66038
n64151
a62264
n60377
a58491
a56604
a54717
a52830
a50943
a49057
a47l70
a45283
a43396
a41509
539623
n37736
a35849
a33962
a32076
a30189
a28302
a26415
524528
«22642
n20755
n18868
a1698]
al5094
a13208
a11321
a09434

Tem .

28.84396
28.90839
29.00507
29.03729
29.06873
29.22559
29.22559
29.3 1973
29.47662
29.53937
29.63229
29.75299
29.84348
29.96420
30.08487
30. 17539
30.1 1502
30.17539
30.20554
30. 14520
30.35639
30.50723
30.68826
30.56757
30.56757
30.86929
3 1.08048
31.20368
3 1. 14081
31.23636
3 1.33441
3 1.43247
3 1.39978
31.30176
31.26904
31.39978
31.43247
31.43247
3 1.56320
31.72342
31.65970
31.56320
31.465 18
31.43247
31.56320
31.75525
31.72342
31.56320
31.30176

r/b Temp. CC)

a07547
a05660
a03774
a01887
.00000
.01887
.03774
.05660
.07547
.09434
.1132]
.13208
.15094
.1698]
.18868
.20755
.22642
.24528
.26415
.28302
.30189
.32076
.33962
.35849
.37736
.39623
.41509
.43396
.45283
.47170
.49057
.50943
.52830
.54717
.56604
.5849]
.60377
.62264
.6415]
.66038
.67925
.6981]
.71698
.73585
.75472
.77359
.79245
.81132
.83019

134

31.33441
3 1. 14081
31.43247
3 1.62787
31.75525
31.94641
3 1.97696
32.00754
3 1.78714
3 1.49783
3 1.72342
3 1.97696
32.06870
32.03812
32.00754
31.94641
32.03812
3 1.81897
31.43247
3 1.59589
31.62787
3 1.33441
31.53052
3 1.56320
3 1.49783
3 1.36710
31.20368
3 l . 17099
31. 17099
31.08048
31.05029
30.95981
30.92963
30.86929
30.77878
30.56757
30.50723
30.47705
30.38657
30.32623
30.29605
30.29605
30.26590
30.] 1502
30.02454
29.87366
29.66248
29.44525
29.28836

1'

.84906
.86792
.88679
.90566
.92453
.94340
.96226
.981 13
1.00000 28.5949]

Tern .

29. 13147
29.00507
28.90839
28.84396
28.90839
28.90839
28.90839
28.78076

Experiment #4

1’

Tem .

-l.00000 24.45111

«98333
«96667
«95000
«93333
«91667
«90000
«88333
«86667
«85000
«83333
«81667
«80000
«78333
«76667
«75000
«73333
«71667
«70000
«68333
«66667
«65000
«63333
«61667
«60000
«58333
«56667
«55000
«53333
«51667
«50000
«48333
«46667
«45000
«43333
«41667
«40000
«38333
«36667
«35000
«33333
«31667
«30000
«28333
«26667
«25000
«23333
«21667
«20000

1214822]
24.451 1 1
24.35779
12126444
2123334
12123334
2138889
24.57556
12176474
12199124
25.21774
25.3 1482
25.21774
25.02362
12192655
12199124
12502362
12502362
12189417
12192655
24.95886
25.05594
2525012
25.31482
25.3 1482
25.34717
2537952
2537952
2537952
25.31482
12534717
25.41 190
12550897
12566452
25.75787
12578897
25.85120
12578897
25.57] 17
2550897
12566452
25.75787
2532675
25.75787
12582010
12588233
26.03787
26J9345

«18333
«16667
«15000
«13333
«11667
«10000
«08333
«06667
«05000
«03333
«01667

.01667
.03333
.05000
.06667
.08333
.10000
.11667
.13333
.15000
.16667
.18333

:21667
.23333

.26667
.28333

.31667
.33333
.35000
.36667
.38333

.41667
.43333
.45000
.46667
.48333

.51667
.53333
.55000
.56667
.58333

.61667

135

Tern

26.22455
26. 19345
26.00678
25.85120
25.85120
25.85120
25.69565
25.66452
25.63342
25.72675
25.82010
25.69565
25.66452
25.72675
25.85 120
26.06897
26.25568
26.25568
26. 10010
25.97565
25.88233
25.94455
26.03787
26.22455
26. 19345
26. 19345
26. 16232
26.19345
26.22455
26. 13 123
25.94455
25.78897
25.69565
25.72675
25.85 120
25.94455
25.91342
25.88233
25.66452
25.50897
25.54CXJ7
25.54(X)7
25.60230
25.54007
25.47659
25.44425
25.47659
25.50897
25.44425

1'

.63333
.65000
.66667
.68333
.70000
.71667
.73333
.75000
.76667
.78333
.80000
.81667
.83333
.85000
.86667
.88333
.90000
.91667
.93333
.95000
.96667
.98333
1.00000 24.7(XJ01

Tern .

25.31482
25.28244
25.25012
25.44425
25.34717
25.25012
25.02362
24.82944
24.86182
24.99124
25. 15302
25.12067
24.95886
24.82944
24.70(X)1
24.66892
24.82944
24.82944
24.66892
24.73236
24.76474
24.73236

Experiment #5

db

Temp. (°C)

-1.00000 27.55979

«98333
«96667
«95000
«93333
«91667
«90000
«88333
«86667
«85000
«83333
«81667
«80000
«78333
«76667
«75000
«73333
«71667
«70000
«68333
«66667
«65000
«63333
«61667
«60000
«58333
«56667
«55000
«53333
«51667
«50000
«48333
«46667
«45000
«43333
«41667
«40000
«38333
«36667
«35000
«33333
«31667
«30000
«28333
«26667
«25000
«23333
«21667
«20000

27.55979
27.59168
27.71942
27.81519
28. l(X)lO
28.22287
28.28427
28. 13077
28.06940
28. 19217
28.40708
28.56055
28.7 1405
28.86878
28.93265
29.02844
29.09094
28.99649
28.99649
29. 18421
29.33969
29.46405
29.52625
29.46405
29.43295
29.49515
29.46405
29.84625
30.23493
30.39688
30.20255
30.04059
29.9 1 104
30.23493
30.42798
30.45908
30.45908
30.52127
30.49017
30.55237
30.67676
30.67676
30.67676
30.67676
30.67676
30.70783
30.77002
30.83222

«18333
«16667
«15000
«13333
«11667
«10000
«08333
«06667
«05000
«03333
«01667

.01667
.03333
.05000
.06667
.08333
. 10000
.11667
.13333
.15000
. 16667
.18333

.21667
.23333
.25000
.26667
.28333

.31667
.33333
.35000
.36667
.38333

.41667
.43333
.45000
.46667
.48333

.51667
.53333
.55000
.56667
.58333

.61667

136

Tern .

31.01877
31.01877
30.86328
30.67676
30.55237
30.58344
30.83222
30.95661
30.86328
30.6 145 3
30.61453
30.61453
30.67676
30.70783
30.73892
30.83222
30.83222
30.801 12
30.64566
30.67676
30.77002
30.89441
3 1.04987
3 1.04987
30.86328
30.70783
30.61453
30.67676
30.73892
30.73892
30.64566
30.58344
30.45908
30.39688
30.52127
30.55237
30.49017
30.49017
30.13776
29.91 104
29.97580
30.10538
30.10538
29.94342
29.81387
29.78146
29.74909
29.71668
29.71668

1'

.63333
.65000
.66667
.68333
.70000
.71667
.73333
.75000
.76667
.78333
.80000
.81667
.83333
.85000
.86667
.88333
.90000
.91667
.93333
.95000
.96667
.98333
1.00000 27.4959]

Tem.

29.74909
29.68430
29.6195 1
29.58844
29.49515
29.37079
29.21530
29.09094
28.86878
28.65268
28.37637
28.25357
28.22287
28.28427
28. 13077
28.03870
27.97482
28.03870
28. 10010
28. 1307 7
27.97482
27.75 134

137

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

C LAST REVISED JUL. 27, 1993 FOR SCOTT HERR

C‘ki‘kit'k'ki*******t************************************************tc

C C
CVCCCCCCCC VARIABLE IDENTIFICATION CCCCCCCC
C C
C C
C*******ti*tttitttttt*tttt****t***********************************C
C C
CDCCCCCCCC DIMENSION BLOCK BLOCK 0000
C C
C C

IMPLICIT REAL*8 (A-H,O-Z)

DIMENSION T(500,5),Y(500),SIGZ(500),B(5),Z(5),A(5),BS(5),
lVINV(5,5),BSS(5),CG(5),BSV(5),R(5,5),EXTRA(20),ERR(500)
1, PS(5,5),P(5,5),PSV(S,5),

1 XTX(5,5),XTY(5),SUM(5),BET(50,2)

CHARACTER*40 DFILE,OUTFIL

C
citii**t*tittt*ttt******t***************titt**********************c
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
ct*tttttt********ii*********************ittt*i***************t**tic
C C
CACCCCCCCC DATA BLOCK BLOCK 0200
C C
DATA EPS,EPSS,IIN,IOUT/1.0D-30,0.00010+0,5,7/
C
ct***ttt******tttt*************************tit********************C
C C
CICCCCCCCC INITIALIZATION BLOCK BLOCK 0400
C C
WRITE(*,*)'ENTER THE NAME OF THE DATA FILE'
READ(*,'(A40)') DFILE
OPEN(8,FILE=DFILE)
WRITE(*,*)'ENTER THE NAME OF THE OUTPUT FILE'
READ(*,'(A40)’) OUTFIL
OPEN(7,FILE=OUTFIL)
C
Ci*t**************************************************************C
C C
CPCCCCCCCC PROCESS BLOCK BLOCK 0500
C C
C --- START INPUT
C BLOCK 1

WRITE17,*)'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 3 MODEL NUMBER, IF SEVERAL MODELS IN SUBROUTINES:
1 MODEL AND SENS'
WRITE(7,*)'IPRINT - 1 FOR USUAL PRINTOUTS, 0 FOR LESS’
WRITE17,*)

IF(N.LE.0) THEN

STOP
END IF
WRITE(*,’(I,9X,"N",8X,"NP",8X,"NT",5X,"ITMAX",5X,
+"MODEL",4X,"IPRINT")’)
WRITE(*,'(7I10)') N,NP,NT,ITMAX,MODL,IPRINT

138

WRITE(7,'(/,9X,"N",8X,"NP",8X,"NT",5X,"ITMAX",5X,

+"MODEL",4X,"IPRINT")')

WRITE(7,'(7IIO)') N,NP,NT,ITMAX,MODL,IPRINT

IOPT=0
--- IF IOPT=O THEN ON THE 2ND AND SUCCEEDING STACKED CASES, THE DATA IS
--- NOT REPRINTED.
IF IPRINT-l, EXTRA PRINT OUT OF ETA, RESIDUALS 8(1),... ARE GIVEN.
BLOCK 2
WRITE(7,*)
WRITE(7,*)’BLOCK 2'
WRITE(7,*)'B(1),B(2),..,B(NP) ARE INITIAL PARAMETER ESTIMATES’
WRITE(7,*)
READ(8,*)(B(I),I=1,NP)
WRITE(7,'(10X,"B(",Il,") - ",E16.5)') (I,B(I),I=1,NP)

0000
I
I
I

DO 150 J1=2,5
BS(J1) = 0
150 CONTINUE

IF(IOPT.LE.0) THEN
C BLOCK 3
WRITE(7,*)
WRITE(7,*)'BLOCK 3’
WRITE(7,*)'J 8 DATA POINT INDEX, Y(J) = MEASURED VALUE'
WRITE(7,*)'SIGMA(J) - STANDARD DEVIATION OF Y(J)'
WRITE(7,*)’T(J,1) - FIRST INDEPENDENT VARIABLE'
WRITE(7,*)
WRITE(7,'(/,9X,”J",6X,"Y(J)",3X,"SIGMA(J)",6X,"T(J,1)"
+.5X."T(J.2)")')
DO 10 I2=1,N
READ(8,*)J,Y(J),SIGZ(J),(T(J,KT),KT=1,NT)
WRITE(7,'(I10,7F10.5)’) J,Y(J),SIGZ(J),(T(J,KT),KT=1,NT)
SIG2(J) - SIG2(J)*SIG2(J)
10 CONTINUE
END IF

C
313 DO 2 IP-1,NP
DO 2 KP=1,NP
PS(KP,IP) = 0
P(KP,IP) = 0
CONTINUE
WRITE(7,'(/,5X,"P(1,KP)",9X,"P(2,KP)",9X,"P(3,KP)",9X,
+“P(4.KP)".9X."P(5.KP>“)')
DO 6 IP=1,NP

READ(8,*)(PS(IP,KP),KP=1,NP)
WRITE(7,'(5D16.5)') (PS(IP,KP),KP=1,NP)

CONTINUE
BLOCK 4
DO 88 IP=1,NP

8 PS(IP,IP)=B(IP)*B(IP)
READ(8,*)IEXTRA
IEXTRA-O FOR NO EXTRA INPUT WHICH COULD BE FOR CONSTANTS
IN MODELS
81 FOR ONE INPUT, NAMELY: EXTRA(I), ETC.
WRITE(7,*)
WRITE(7,*)'BLOCK 4'
WRITE(7,*)'IEXTRA - NO. OF EXTRA(I) PARAMETERS, 0 IF NONE'
WRITE(7,*)
WRITE(7,'(10X,"IEXTRA - ",IlO)')IEXTRA
IF(IEXTRA .LT. 1) GOTO 21
WRITE(7,*)
WRITE(7,*)'BLOCK 5'
WRITE(7,*)'EXTRA(1),... ARE EXTRA CONSTANTS USED AS DESIRED’
WRITE(7,*)
READ(8,*)(EXTRA(IE),IE=1,IEXTRA)
WRITE(7,'("EXTRA(",I2,") - ",F16.5)') (IE,EXTRA(IE),IE=1
1,IEXTRA)

21 CONTINUE

m OQOOOOOON

000

C

C --- ADD BLANK CARD AFTER LAST INPUT CARD

C ---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
1 BE SMALLER THAN SY'
WRITE(7,*)' SYP DECREASES TOWARD A POSITIVE CONSTANT’
WRITE(7,*)'G = MEASURE OF THE SLOPE, SHOULD BECOME SMALLER AS
lITERATIONS PROCEED'

18

19

99

c _--

CCCC

20

29

30

4o

50

41

51
52

139

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,*)
WRITE(7,*)
DO 18 IL=1,NP
BS(IL)-B(IL)
CG(IL) - 0
CONTINUE
DO 19 IP=1,NP
XTY(IP)=0.0D+0
DO 19 KP=1,NP
P(KP,IP) = PS(KP,IP)
XTX(IP,KP)=0.0D+0
CONTINUE
I - 0
MAX - 0

MAX = MAX + l
START BASIC LOOP GIVES B(I) AND SY

SY - 0.0D+0
DO 100 13=1,N
I - I3
CALL MODEL
CALL SENS
CALL MODEL
RISD - Y(I)-ETA
SY - SY + RISD*RISD/SIGZ(I)
SUMX- 0.0D+O
DO 20 K=I,NP
XTY(K)-XTY(K)+Z(K)*RISD/$IG2(I)
DO 20 L=1,NP
SUMX= SUMX+ Z(L)*P(K,L)*Z(K)
XTx(K,L)= XTX(K,L) + Z(L)*Z(K)/SIGZ(I)
CONTINUE
DELTA - SIGZ(I) + SUMX
DO 29 JJ=1,NP
A(JJ) = 0.0D+0
CONTINUE
DO 30 JA=1,NP
DO 30 KA-1,NP
A(JA) - A(JA) + Z(KA)*P(JA,KA)
CONTINUE
CS - 0.0D+0
DO 40 JC=1,NP
CS - CS + Z(JC)*(B(JC)-BS(JC))
CG(JC) s CG(JC) + 2(JC)*RISD/SIGZ(I)
CONTINUE
C - Y(I) - CS - ETA
DO 50 IB-1,NP
B(IB) - B(IB) + (A(IB)*C)/DELTA
CONTINUE
DO 41 ISV=1,NP
DO 41 JSV-1,NP
PSV(JSV,ISV) - P(JSV,ISV)
CONTINUE
DO 52 Iv-1,NP
DO 52 IU=Iv,NP
SUMP - 0.0D+0
DO 51 KP=1,NP
DO 51 JP=1,NP
IF(KP-IV.EQ.0.0R.JP-IU.EQ.0) GOTO 51
P501 - PSV(KP,JP)*PSV(IU,IV)
P802 - PSV(IU,KP)*PSV(IV,JP)
PSQ = PSQl - PSQZ
IF(DABS(PSQl)+DABS(PSQZ).LT.1.D-15) THEN
RP = PSQ * 1.D15
ELSE
RP - PSQ / (DABS(PSQl)+DABS(PSQZ))
END IF
RP a ABS(RP)
RPP = RP - 1.00-12
IF(RPP.LE.0.0D+O) THEN
PSQ - 0.0D+0
END IF
SUMP = SUMP + Z(JP)*Z(KP)*PSQ
CONTINUE
P(IU,IV) = (P$V(IU,IV)*SIGZ(I)+SUMP)/DELTA
CONTINUE

140

D0 53 IV=2,NP
IVM = IV - 1
DO 53 IU - 1,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,"I",6X,"ETA",3X,"RES.",2X,
lllB(1)II'7X'IOB(2)'0’7X'IIB(3)II'7x’llB(4)00)!)

END IF

C WRITE(7,'(I4,6E12.5)')I,ETA,RISD,(B(JC),JC=1,NP)
WRITE(7,'(I3,F10.2,F8.3,5E11.4)')I,ETA,RISD,(B(JC),JC=1,NP)

END IF

100 CONTINUE

C --- END BASIC LOOP, GIVES B(I) AND SY

C --- START BOX-KANEMASU MODIFICATION

C

C START BOX-KANEMASU MODIFICATION

IF(MAX-1)104,104,103
103 ss-SY/2.0D+0
IF(ss-SYP)104,104,105
105 DO 210 IBS=1,NP
B(IBS)= BSV(IBS)
210 CONTINUE
WRITE(IOUT,212)
212 FORMAT(7X,'USE BSV(IBS)’)
GOTO 211
104 CONTINUE
DO 102 IBS=1,NP
BSS(IBS)= BS(IBS)
102 CONTINUE
ALPHA= 2.0D+0
AA= 1.1D+0
110 ALPHA= ALPHA/2.0D+0
DO 116 IBs-1,NP
BS(IBS)- BSS(IBS) + ALPHA*( B(IBS)-BSS(IBS) )
BSV(IBS)= BS(IBS)
116 CONTINUE
INDEX-0
G= 0.0D+O
DO 115 IP=1,NP
DELB= BS(IP)-BSS(IP)
G- G + DELB*CG(IP)
RATIO- DELB/( BSS(IP)+EPS )
RATIO= ABS(RATIO)
IF(RATIO-EPSS)113,113,114

113 INDEX= INDEX+1
WRITE(IOUT,314)

314 FORMAT(7X,’MAX',8X,’NP’,5X,'INDEX',8X,’IP’)
WRITE(7,'(7IlO)') MAX,NP,INDEX,IP

114 CONTINUE

C WRITE(7,122) I,Y(I),ETA,RISD,Z(IP),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/SIGZ(I)
117 CONTINUE
IF(NP-INDEX)106,106,107
106 H=1.0D+0
GOTO 132
107 CONTINUE
SYN= SYP*0.999D+0
IF(SYN-SY)112,112,111
111 IF(ALPHA-0.01D+0)109,109,110
109 WRITE(7,108) ALPHA,SYP,SY
108 FORMAT(3X,'ALPHA TOO SMALL,ALPHA=',F12.6,2X,'SYP=',E15.6,2X,
l'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

141

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+O*ALPHA*G )
GOTO 132

131 CONTINUE
Ha 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',10X,'H',13X,’G',12X,

1'5Y’,11X,'SYP')

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

122 FORMAT(I8,1F13.6,4E14.6)
WRITE(7,'(10X,"B(",Il,") - ",E16.6)') (I,B(I),I=1,NP)
WRITE(*,'(10X,"B(",Il,") = ",E16.6)') (I,B(I),I=1,NP)

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

1"P(4,KP)",9X,"P(S,KP)")')
DO 206 IP=1,NP
WRITE(7,207) (P(IP,KP),KP=1,NP)
206 CONTINUE
207 FORMAT(5015.7)
WRITE(7,135)
135 FORMAT(5X,'CORRELATION MATRIX')
DO 136 IR=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 137 IR-1,NP
WRITE(7,'(5E15.7)') (R(IR,III),III=1,IR)

137 CONTINUE

DO 126 IPS=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,'(SE15.7)’)(XTX(K,III),III=1,NP)
WRITE(7,*)'XTY(I),I=1,NP, WHERE Y IS RESID’
WRITE(7,’(5E15.7)')(XTY(I),I=1,NP)

127 FORMAT(3X,'IPS=',IQ,3X,’PS(IPS,IPS)=',DlS.8)
WRITE(7,*)'XTY(I),I=1,NP, Y IS Y, NOT RESID'
WRITE(7,'(5E15.7)')(SUM(I),I=1,NP)

Do 119 IP=1,NP
XTY(IP)=0.0D+0
DO 119 KP=1,NP
P(IP.KP)- PS(IP,KP)
XTX(IP,KP)=0.0D+0

119 CONTINUE

DO 120 IP=1,NP
BS(IP)= B(IP)
CG(IP)= 0.0D+0

120 CONTINUE
WRITE(7,314)

WRITE(7,'(7I10,4F10.4)') MAX,NP,INDEX,IP
IF(NP-INDEX)101,101,123

123 CONTINUE
M=ITMAX
IF(MAX-M)99,99,101

101 CONTINUE
IF(IPRINT)133,133,134

133 IPRINT=IPRINT+1
GOTO 99

134 CONTINUE

C

1000 CONTINUE
CLOSE(IIN)
CLOSE (IOUT)

C
C********************************************‘k‘k*‘k‘k**************‘k*c

C C
CECCCCCCCC ERROR MESSAGES BLOCK 0900
C C

C C

142

C********i**itiiiit*******ii*i************t***********************C

C C
CFCCCCCCCC FORMAT STATEMENTS BLOCK 9000
C C
C C

c*******t**********iiii**********t***************************i****C

C
STOP
END
SUBROUTINE MODEL
C THIS SUBROUTINE IS FOR CALCULATING ETA, THE MODEL VALUE
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION T(500,5),Y(500),SIGZ(500),B(5),Z(5),BET(50,2),
+A(5),BS(5),VINV(5,5),EXTRA(20)
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

C WRITTEN BY JAMES V. BECK
PI=4.0D+0*DATAN(1.0D+0)
C IF(MODL .EQ. 1) GOTO 800

800 CONTINUE
TR=T(I,1)/EXTRA(2)
ETA=BS(1)+BS(2)*EXTRA(1)*(1.0D+0-TR*TR)
1000 CONTINUE

C WRITE(*,*)'I,T(I,1),ETA,Z(1)',I,T(I,1),ETA,Z(1)
RETURN
END
SUBROUTINE SENS
C THIS SUBROUTINE IS FOR CALCULATING THE SENSITIVITY COEFFICIENTS

IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION T(500,5),Y(500),SIGZ(500),B(5),BET(50,2),
+Z(5),A(5),BS(5),VINV(5,5),EXTRA(20)
DIMENSION P(S,S),PS(5,5),SUM(5)
COMMON T,N,2,85,I,ETA,PS,P,B,A,Y,SIGZ,MODL,VINV,NP,EXTRA
COMMON/MOD/AA,TL,SUM,BET,IH,CONST
PI=4.0D+0*DATAN(1.0D+O)
z(1)=1.0D+0
TR=T(I,1)/EXTRA(2)
Z(2)=EXTRA(1)*(l.0D+0-TR*TR)
DO 312 IPP=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(JPP)=SUM(JPP)+Z(JPP)*(Y(I)-TZ)/SIG2(I)
2000 CONTINUE

RETURN

END

143

Appendix D.

NLIN Output File for Experiment #5

BEGIN LISTING INPUT QUANTITIES

BLOCK 1

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

NT I NO. OF INDEPENDENT VARIABLES

ITMAX I MAXIMUM NO. OF ITERATIONS

MODEL I MODEL NUMBER, IF SEVERAL MODELS IN SUBROUTINES: MODEL AND SENS
IPRINT I 1 FOR USUAL PRINTOUTS, 0 FOR LESS

N NP NT ITMAX MODEL IPRINT
121 2 1 5 l 1
BLOCK 2
B(1),B(2),..,B(NP) ARE INITIAL PARAMETER ESTIMATES
B(1) = .27SOOE+02
8(2) = .40000E-02
BLOCK 3
J = DATA POINT INDEX, Y(J) I MEASURED VALUE
SIGMA(J) = STANDARD DEVIATION OF Y(J)
T(J,1) I FIRST INDEPENDENT VARIABLE
J Y(J) SIGMA(J) T(J,1) T(J,2)
1 27.55979 1.00000 -1.00000
2 27.55979 1.00000 -.98333
3 27.59168 1.00000 -.96667
4 27.71942 1.00000 -.95000
5 27.81519 1.00000 -.93333
6 28.10010 1.00000 -.91667
7 28.22287 1.00000 -.90000
8 28.28427 1.00000 -.88333
9 28.13077 1.00000 -.86667
10 28.06940 1.00000 -.85000
11 28.19217 1.00000 —.83333
12 28.40708 1.00000 -.81667
13 28.56055 1.00000 -.80000
14 28.71405 1.00000 -.78333
15 28.86878 1.00000 -.76667
16 28.93265 1.00000 -.75000
17 29.02844 1.00000 -.73333
18 29.09094 1.00000 -.71667
19 28.99649 1.00000 -.70000
20 28.99649 1.00000 -.68333
21 29.18421 1.00000 -.66667
22 29.33969 1.00000 -.65000
23 29.46405 1.00000 -.63333
24 29.52625 1.00000 -.61667
25 29.46405 1.00000 -.60000
26 29.43295 1.00000 -.58333
27 29.49515 1.00000 -.56667
28 29.46405 1.00000 -.55000
29 29.84625 1.00000 -.53333
30 30.23493 1.00000 -.51667
31 30.39688 1.00000 -.50000
32 30.20255 1.00000 -.48333
33 30.04059 1.00000 -.46667
34 29.91104 1.00000 -.45000
35 30.23493 1.00000 -.43333
36 30.42798 1.00000 -.41667
37 30.45908 1.00000 -.40000
38 30.45908 1.00000 -.38333
39 30.52127 1.00000 -.36667
40 30.49017 1.00000 -.35000
41 30.55237 1.00000 -.33333
42 30.67676 1.00000 -.31667
43 30.67676 1.00000 -.30000
44 30.67676 1.00000 -.28333

1
1
l
1
1
1
1
l
1
1
l
1
1
1
1
1

BLOCK 4

IEXTRA I NO. OF EXTRA(I) PARAMETERS,

06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21

30.67676
30.67676
30.70783
30.77002
30.83222
31.01877
31.01877
30.86328
30.67676
30.55237
30.58344
30.83222
30.95661
30.86328
30.61453
30.61453
30.61453
30.67676
30.70783
30.73892
30.83222
30.83222
30.80112
30.64566
30.67676
30.77002
30.89441
31.04987
31.04987
30.86328
30.70783
30.61453
30.67676
30.73892
30.73892
30.64566
30.58344
30.45908
30.39688
30.52127
30.55237
30.49017
30.49017
30.13776
29.91104
29.97580
30.10538
30.10538
29.94342
29.81387
29.78146
29.74909
29.71668
29.71668
29.74909
29.68430
29.61951
29.58844
29.49515
29.37079
29.21530
29.09094
28.86878
28.65268
28.37637
28.25357
28.22287
28.28427
28.13077
28.03870
27.97482
28.03870
28.10010
28.13077
27.97482
27.75134
27.49591

1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000

144

-.26667
-.25000
-.23333
-.21667
-.20000
-.18333
-.16667
-.15000
-.13333
-.11667
.10000
.08333
.06667
.05000
.03333

.71667
.73333
.75000
.76667
.78333
.80000
.81667
.83333
.85000
.86667
.88333
.90000
.91667
.93333
.95000
.96667
.98333
1.00000

0 IF NONE

145

IEXTRA = 2

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

EXTRA( 1) I 831.86000
EXTRA( 2) I 1.00000
END INPUT QUANTITIES - - BEGIN OUTPUT CALCULATIONS

SY = SUM OF SQUARES FOR PRESENT PARAMETER VALUES
SYP I SUM OF SQUARES FOR GAUSS PARAMETER VALUES, SHOULD BE SMALLER THAN SY
SYP DECREASES TOWARD A POSITIVE CONSTANT

G I 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

I ETA RES. 8(1) 8(2) 8(3) 8(4)
1 27.50 .060 .2756E+02 .4000E-02
2 27.61 -.050 .2751E+02 .39768-02
3 27.72 -.126 .2746E+02 .3921E-02
4 27.82 -.105 .2745E+02 .3883E-02
5 27.93 -.114 .2744E+02 .3844E-02
6 28.03 .069 .2746E+02 .39748-02
7 28.13 .091 .2747E+02 .4105E-02
8 28.23 .053 .2747E+02 .4171E-02
9 28.33 -.197 .2747E+02 .3973E-02
10 28.42 -.354 .2747E+02 .3669E-02
11 28.52 -.325 .2748E+02 .3473E-02
12 28.61 -.201 .2748E+02 .34448-02
13 28.70 -.137 .274BE+02 .34858-02
14 28.79 -.072 .2747E+02 .35748-02
15 28.87 -.003 .27468+02 .3698E-02
16 28.96 -.023 .274SE+02 .3782E-02
17 29.04 -.010 .2744E+02 .3857E-02
18 29.12 -.027 .2744E+02 .3906E-02
19 29.20 -.201 .2744E+02 .3854E-02
20 29.27 -.277 .274SE+02 .3777E-02
21 29.35 -.164 .27468+02 .37708-02
22 29.42 -.082 .2745E+02 .38008-02
23 29.49 -.029 .2744E+02 .3848E—02
24 29.56 -.O36 .2744E+02 .38848-02
25 29.63 -.166 .2744E+02 .38708-02
26 29.70 -.262 .2745E+02 .3827E-02
27 29.76 -.264 .2745E+02 .3792E-02
28 29.82 -.357 .2747E+02 .3737E-02
29 29.88 -.035 .2746E+02 .378OE-02
30 29.94 .296 .2743E+02 .3901E—02
31 30.00 .401 .2740E+02 .4031E-02
32 30.05 .152 .2739E+02 .4084E-02
33 30.10 -.062 .2739E+02 .40828-02
34 30.15 -.243 .274OE+02 .40428-02
35 30.20 .032 .2740E+02 .4062E-02
36 30.25 .178 .2739E+02 .41078-02
37 30.30 .164 .2738E+02 .4144E-02
38 30.34 .121 .2737E+02 .4168E-02
39 30.38 .141 .2736E+02 .41928-02
40 30.42 .070 .2736E+02 .4203E-02
41 30.46 .095 .27368+02 .421SE-02
42 30.49 .183 .2735E+02 .4239E-02
43 30.53 .149 .273SE+02 .42558-02
44 30.56 .116 .2735E+02 .4264E-02
45 30.59 .086 .2734E+02 .4269E-02
46 30.62 .057 .2734E+02 .4270E-02
47 30.65 .062 .2734E+02 .4270E—02
48 30.67 .099 .2734E+02 .427SE-02
49 30.69 .138 .2734E+02 .4283E-02
50 30.72 .303 .2733E+02 .43088-02
51 30.74 .284 .2733E+02 .43298-02
52 30.75 .111 .2733E+02 .433OE-02
53 30.77 -.092 .2733E+02 .4313E-02
54 30.78 -.230 .2734E+02 .4284E-02
55 30.79 -.211 .2735E+02 .4259E-02
56 30.80 .028 .273SE+02 .42568-02
57 30.81 .144 .2735E+02 .4263E-02

58
59
60
61
62
63

65
66
67
68

70
71
72
73
74
75
76
77
78
79
80

82
83
84

86
87

89
90
91
92
93
94
95

97

98

99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121

.4726766D-01
-.70922OSD-04
CORRELATION MATRIX

30.82 .044
30.82 - 209
30.83 -.212
30.83 -.213
30.83 - 150
30.82 -.116
30.82 -.080
30.81 .020
30.80 .028
30.79 .007
30.78 -.136
30.77 -.092
30.75 .017
30.74 .159
30.72 .334
30.69 .356
30.67 .192
30.65 .062
30.62 -.005
30.59 .086
30.56 .179
30.53 211
30.49 .152
30.46 .126
30.42 .039
30.38 .017
30.34 .183
30.30 .257
30.25 .240
30.20 .288
30.15 - 016
30.10 -.192
30.05 -.074
30.00 .110
29.94 .166
29.88 .062
29.82 -.007
29.76 022
29.70 .054
29.63 .087
29.56 .155
29.49 .256
29.42 .263
29.35 .271
29.27 .315
29.20 .298
29.12 .252
29.04 .177
28.96 .135
28.87 -.003
28.79 -.133
28.70 - 322
28.61 -.355
28.52 -.294
28.42 -.139
28.33 -.197
28.23 -.192
28.13 -.157
28.03 .007
27.93 .171
27.82 .306
27.72 .257
27.61 .141
27.50 -.004
MAX H
1 1.006253
8(1) I
8(2) =
P(1,KP)

-.70922050-04
.1289619D-06

.1000000E+01
-.9083821E+00 .1000000E+01
XTX(I,K),K=1,NP
.1210000E+03 .6654418E+05
.6654418E+05 .4428743E+08

XTY(I),I=1,NP, WHERE Y IS RESID

.2434174E+01

.2460668E+04

146

.2735E+02 .4261E-02
.2735E+02 .4240E-02
.2736E+02 .4221E-02
.2737E+02 .4203E-02
.2737E+02 .4190E-02
.2737E+02 .4181E-02
.2737E+02 .4174E-02
.2737E+02 .41748-02
.2737E+02 .41758-02
.2737E+02 .4174E-02
.27388+02 .4166E-02
.2738E+02 .41608-02
.2738E+02 .41618-02
.2738E+02 .4168E-02
.2737E+02 .4183E-02
.2737E+02 .4197E-02
.2737E+02 .42048-02
.2737E+02 .4205E-02
.2737E+02 .4204E-02
.2737E+02 .4206E-02
.2737E+02 .42118-02
.27368+02 .4217E-02
.27368+02 .4221E-02
.27368+02 .4223E-02
.2736E+02 .4223E-02
.27368+02 .4223E-02
.2736E+02 .42268-02
.2736E+02 .4230E-02
.27368+02 .4233E-02
.2737E+02 .4236E-02
.2737E+02 .42368-02
.27368+02 .42358-02
.273GE+02 .4234E-02
.2736E+02 .4234E-02
.2737E+02 .42348—02
.2737E+02 .4234E-02
.2737E+02 .4234E-02
.2737E+02 .4234E-02
.2737E+02 .4233E-02
.2737E+02 .42318-02
.2737E+02 .42288-02
.2738E+02 .42218-02
.2739E+02 .42148-02
.2740E+02 .42068-02
.2741E+02 .41958-02
.274ZE+02 .4183E-02
.2742E+02 .4173E-02
.2743E+02 .4165E-02
.2744E+02 .41588-02
.2744E+02 .4158E-02
.2743E+02 .41648-02
.274ZE+02 .41818-02
.2741E+02 .4199E-02
.2739E+02 .42148-02
.2739E+02 .4219E-02
.2738E+02 .4227E-02
.2738E+02 .4235E-02
.273SE+02 .42408-02
.273BE+02 .42338-02
.2739E+02 .4215E-02
.274lE+02 .4187E-02
.2743E+02 .4164E-02
.2744E+02 .4149E-02
.2744E+02 .4145E-02
G SY
.211317E+00 .387968E+01
.274402E+02
.414560E-02
P(2.KP) P(3oKP)

SYP
.366705E+01

P(4.KP)

P(5,KP)

XTY(I),I=1,NP, Y IS Y, NOT RESID
-.409000OE-02

SEQUENTIAL ESTIMATES OF THE PARAMETERS GIVEN BELOW

HFJFIHrJFJH
(hUhthNFACDwCD~JanhHAN)H

I

MAX
1

ETA
27.44
27.55
27.67
27.78
27.88
27.99
28.10
28.20
28.30
28.40
28.49
28.59
28.68
28.77
28.86
28.95
29.03
29.12
29.20
29.28
29.36
29.43
29.51
29.58
29.65
29.72
29.78
29.85
29.91
29.97
30.03
30.08
30.14
30.19
30.24
30.29
30.34
30.38
30.43
30.47
30.51
30.54
30.58
30.61
30.64
30.67
30.70
30.73
30.75
30.77
30.79
30.81
30.83
30.84
30.85
30.86
30.87
30.88
30.88
30.89
30.89
30.89
30.88
30.88
30.87
30.86
30.85
30.84
30.83
30.81
30.79

.0000000E+00
NP INDEX
2 0

RES.
.120
.006

-.075

-.057

-.069
.109
.127
.086

-.168

-.328

-.302

-.182

-.121

-.059
.007

-.016

-.006

-.027

-.202

-.282

—.172

-.092

-.041

-.051

-.183

-.282

-.286

-.381

-.062
.267
.370
.119

-.097

-.279

-.006
.138
.122
.077
.096
.024
.047
.134
.098
.065
.033
.004
.007
.043
.081
.246
.226
.052

-.151

-.289

-.271

-.033
.083

-.017

-.270

-.273

-.274

-.211

-.177

—.141

-.041

-.033

-.053

-.196

-.151

-.041
.101

8(1)

.2756E+02
.27563+02
.27SSE+02
.27538+02
.2751E+02
.2747E+02
.2744E+02
.2743E+02
.2747E+02
.2752E+02
.2754E+02
.2753E+02
.27SZE+02
.27SOE+02
.27488+02
.274GE+02
.274SE+02
.2744E+02
.274SE+02
.27468+02
.2746E+02
.27462+02
.27458+02
.2744E+02
.2744E+02
.27453+02
.27463+02
.2747E+02
.274GE+02
.2743E+02
.27403+02
.273QE+02
.2739E+02
.274OE+02
.27408+02
.273SE+02
.27388+02
.2737E+02
.27362+02
.27368+02
.2736E+02
.273SE+02
.2734E+02
.2734E+02
.273dE+02
.2734E+02
.27348+02
.2734E+02
.2734E+02
.2733E+02
.27323+02
.273ZE+02
.2733E+02
.2734E+02
.273SE+02
.273SE+02
.2734B+02
.2734E+02
.273SE+02
.2736E+02
.27368+02
.2737E+02
.2737E+02
.2737E+02
.2737E+02
.2737E+02
.2737E+02
.273BE+02
.27388+02
.27388+02
.273SE+02

147

IP
3

8(2)

.4146E-02
.8494E-05
.58508-03
.1882E-02
.2492E-02
.3844E-02
.4442E-02
.45468-02
.3930E-02
.3288E-02
.3009E-02
.3061E-02
.32018-02
.3383E-02
.35858-02
.3714E-02
.3819E-02
.3885E-02
.38258-02
.3739E-02
.3735E-02
.3774E-02
.3830E-02
.3872E-02
.3857E-02
.3812E-02
.3775E-02
.3718E—02
.3765E-02
.3895E-02
.40338-02
.4088E-02
.4086E-02
.4044E-02
.4065E-02
.4112E-02
.4149E-02
.4174E-02
.4199E-02
.4209E-02
.4222E-02
.42468-02
.4262E-02
.4272E-02
.4276E-02
.4276E-02
.4277E-02
.42818-02
.4290E-02
.43158-02
.43358-02
.4337E-02
.4319E-02
.4289E-02
.4264E-02
.4261E-02
.4267E-02
.4266E-02
.4244E-02
.4225E-02
.4206E-02
.4193E-02
.41838-02
.4177E-02
.4177E-02
.4177E-02
.4177E-02
.4168E-02
.41638-02
.4163E-02
.4170E-02

8(3)

8(4)

XTY(I),I=1,NP, WHERE Y IS RESID

-.1530085E-01 -.6287375E+01

XTY(I),I=1,NP, Y IS Y, NOT RESID

.55739718-01 .0000000E+00
MAX NP

2 2 2

INDEX

148

72 30.77 .277 .2737E+02 .4185E-02
73 30.75 .299 .2737E+02 .4200E-02
74 30.73 .136 .2737E+02 .4207E-02
75 30.70 .007 .2736E+02 .4208E-02
76 30.67 -.059 .2737E+02 .4207E-02
77 30.64 .033 .273GE+02 .4209E-02
78 30.61 .127 .27368+02 .4214E-02
79 30.58 .161 .27368+02 .42208-02
80 30.54 .103 .2736E+02 .4223E-02
81 30.51 .078 .273GE+02 .42268-02
82 30.47 -.007 .2736E+02 .4226E-02
83 30.43 -.028 .2736E+02 .4226E-02
84 30.38 .139 .2736E+02 .4229E-02
85 30.34 .215 .2736E+02 .42338-02
86 30.29 .200 .2736E+02 .4236E-02
87 30.24 .249 .2736E+02 .4239E-02
88 30.19 -.053 .2736E+02 .4239E-02
89 30.14 -.227 .27368+02 .4238E-02
90 30.08 -.107 .2736E+02 .4237E-02
91 30.03 .079 .27368+02 .4237E-02
92 29.97 .137 .27368+02 .4237E-02
93 29.91 .036 .2736E+02 .4237E-02
94 29.85 —.032 .273SE+02 .4237E-02
95 29.78 .000 .2736E+02 .4237E-02
96 29.72 .034 .2737E+02 .4236E-02
97 29.65 .069 .2737E+02 .4234E-02
98 29.58 .139 .2737E+02 .4231E-02
99 29.51 .244 .2738E+02 .4224E-02
100 29.43 .253 .2739E+02 .4217E-02
101 29.36 .263 .2739E+02 .4208E-02
102 29.28 .310 .274OE+02 .4197E-02
103 29.20 .296 .2741E+02 .4186E-02
104 29.12 .253 .274ZE+02 .41758-02
105 29.03 .181 .2743E+02 .4167E-02
106 28.95 .142 .2743E+02 .416OE-02
107 28.86 .007 .2744E+02 .4160E-02
108 28.77 -.120 .2743E+02 .4166E—02
109 28.68 -.305 .274ZE+02 .4183E-02
110 28.59 -.335 .2740E+02 .4201E-02
111 28.49 -.271 .2739E+02 .4216E-02
112 28.40 -.113 .2739E+02 .4221E-02
113 28.30 -.168 .2738E+02 .42308-02
114 28.20 -.159 .273BE+02 .4237E-02
115 28.10 -.121 .2737E+02 .4242E-02
116 27.99 .048 .2738E+02 .42358-02
117 27.88 .215 .2739E+02 .4217E-02
118 27.78 .354 .274lE+02 .4189E-02
119 27.67 .309 .2743E+02 .4165E-02
120 27.55 .197 .2744E+02 .4150E-02
121 27.44 .056 .2744E+02 .41468-02
MAX NP INDEX IP
2 2 1 1
MAX NP INDEX IP
2 2 2 2
H G
1.000000 .2523258-05
8(1) .274399E+02
8(2) I .4145888-02
P(1.KP) P(2.KP) P(3.KP)
.4758757D-01 -.7150281D-04
-.71502810-04 .1300165D-06
CORRELATION MATRIX
.1000000E+01
-.90902788+00 .1000000E+01
XTX(I,K),K=1,NP
.1210000E+03 .6654418E+05
.6654418E+05 .4428743E+08

SY
.366704E+01

SYP
.366704E+01

P(4.KP)

P(5,KP)