November 2001 

i

NIST Recommended Practice Guide

Special Publication 960-7

Capacitance Cell

Measurement of the

Out-of-Plane Expansion

of Thin Films

Chad R. Snyder

Frederick I. Mopsik

Materials Science and

Engineering Laboratory

November 2001

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director
ii

◆ Capacitance Cell Measurement

Certain commercial entities, equipment, or materials may be identified in

this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or

endorsement by the National Institute of Standards and Technology, nor is it

intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology

Special Publication 960-7

Natl. Inst. Stand. Technol.

Spec. Publ. 960-7

44 pages (November 2001)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2003

For sale by the Superintendent of Documents

U.S. Government Printing Office

Internet: bookstore.gpo.gov Phone: (202) 512–1800 Fax: (202) 512–2250

Mail: Stop SSOP, Washington, DC 20402-0001
iii

FOREWORD

The coefficient of thermal expansion (CTE) continues to be a critical

parameter in modeling for the microelectronics industry. Over the past several

years, NIST has worked to develop a robust and highly-accurate means for

determining the CTE of films with thicknesses ranging from 2 μm up to 1 cm.

To make this metrology accessible to a wider community and to help users to

adhere to “good practice” procedures when using this metrology, this Guide

was created. At the end of this Guide, the data reduction technique necessary

for using this metrology for the determination of hygrothermal expansion is

discussed, and a detailed look at the uncertainties associated with this

metrology is provided.

More information on the SP 960 series can be found on the Internet at

http://www.nist.gov/practiceguides. This website includes a complete

list of NIST Practice Guides and ordering information.

Foreword ◆
iv

◆ Capacitance Cell Measurement
v

Acknowledgments ◆

ACKNOWLEDGMENTS

The authors would like to thank the NIST Office of Microelectronics

Programs for financial support in the development of this metrology.
vi

◆ Capacitance Cell Measurement
vii

Table of Contents ◆

TABLE OF CONTENTS

List of Figures .....................................................................................ix

1. Introduction......................................................................................1

2. Scope................................................................................................3

3. Test Specimens ................................................................................4

3.1 Size..........................................................................................4

3.2 Condition .................................................................................4

3.3 Preparation ..............................................................................4

4. Apparatus .........................................................................................5

4.1 Instrumentation .......................................................................5

4.2 Capacitance Cell .....................................................................6

4.2.1 Electrode Construction............................................................6

4.2.2 Holder Construction ................................................................9

4.2.3 Cell Assembly .........................................................................9

4.3 Accessories ..........................................................................14

4.3.1 Coaxial Cable ........................................................................14

5. Procedure .......................................................................................17

5.0a Thermal Expansion ...............................................................17

5.0b Hygrothermal Expansion ......................................................18

5.1 Calibration .............................................................................18

5.2 Test Measurements ..............................................................20

5.3 Evaluation ..............................................................................20

5.4 Calculation of Coefficient of Thermal Expansion ..................22

5.5 Report of CTE Data ..............................................................22

5.6 Plot of CTE Data ...................................................................23
viii

◆ Capacitance Cell Measurement

6. Notes ...............................................................................................24

6.4 “Debugging” a Reproducibility Problem ................................24

APPENDIX A — Dielectric Constant of Air .....................................25

APPENDIX B — Discussion of Uncertainties .................................27

B.1. Electrode Tilt .........................................................................27

B.2. Temperature and Pressure Uncertainties .............................27

B.3. Thickness Dependence of the Effective Area ......................28

B.4. Other Sources of Uncertainty with Respect to

Thickness Reproducibility .....................................................29

B.5. Capacitance Bridge Calibration .............................................30

References ..........................................................................................31
ix

List of Figures

Figure 1. Fused quartz blank electrode construction. Note that the

drawing is not to scale; the supplied measurements should

be used. (The diameter of the blank is 10 cm as specified

in the text.)....................................................................................... 7

Figure 2. Electrode construction showing the nichrome coating,

placement of the copper wires, conducting paint, and

one of the three screw clamp holes. Note that the

drawing is not to scale; the supplied measurements

should be used.................................................................................. 8

Figure 3. Cross-sectional view of the dimensions for construction

of the electrode holder. The materials for the cell holder

are brass or copper. Note that the drawing is not to scale;

the supplied measurements should be used ................................... 10

Figure 4. Top and bottom views of the electrode holder. The materials

for the cell holder are brass or copper. Note that the

drawing is not to scale; the supplied measurements

should be used................................................................................. 11

Figure 5. Side and bottom views of the dimensions for construction

of the electrode holder’s bottom cover plate. Note that

the drawing is not to scale; the supplied measurements

should be used................................................................................ 12

Figure 6. Side and bottom views of the dimensions for construction

of the aluminum shield plate. Note that the drawing is

not to scale; the supplied measurements should be used ............... 13

Figure 7. Schematic of electrical connections to bottom electrode .............. 15

Figure 8. Diagram of assembled capacitance cell ........................................ 16

Figure 9. Typical thickness versus time plot for a material that

undergoes dimensional contraction (creep) under load

after a temperature step. (This plot is given to aid the

reader only; it should not be interpreted as actual data.) .............. 21

Table of Contents ◆
x

◆ Capacitance Cell Measurement

Figure 10. Typical thickness versus time plot for a material that

does not undergo dimensional contraction (creep) under

load after a temperature step. (The plot is given to aid the

reader only; it should not be interpreted as actual data.) .............. 21

Figure 11. Typical relative expansion curve. (The plot is given to

aid the reader only; it should not be interpreted as

actual data.) ................................................................................... 23
1

Introduction ◆

1. INTRODUCTION

The coefficient of thermal expansion (CTE) is one of the key design

parameters in the microelectronics industry. CTE mismatch between different

materials is a prime concern to this industry since thermal stresses can build up

as a result and lead to device failure. To minimize the chance of such failure,

it is vitally important that reliable values for the CTE of the materials be known

so that modeling can be performed to assist design of the structure. CTE

values for these materials are often obtained from measurements performed

on large (thick) samples because of the limitations in sensitivity of most

commercial devices. As a result, most standard test methods require samples

thicker than 1 mm.[1,2] Unfortunately, many of these materials can be formed

only as thin layers, and there is some question in the literature regarding the

thickness dependence of the CTE.[3ñ5]

To improve upon this situation, a metrology to obtain CTE values was

developed that was based on capacitance measurements ó with a resulting

wide range of measurable thickness and a high sensitivity. (Throughout this

guide, this metrology will be referred to as a ìcapacitance cell.î) The

capacitance-based technique was chosen since commercial capacitance

bridges exist that can measure capacitance to a relative standard uncertainty

of better than 10ñ6 pF/pF. With proper design of the measurement cell, the

relative uncertainty in thickness can approach the relative uncertainty in

capacitance. Many researchers have recognized this fact and a large number

of different designs have been proposed.[6ñ9] The uniqueness of the metrology

in this guide relative to the previous capacitance-based metrologies lies in

the fact that the measurement of expansion is completely independent of the

dielectric properties of the material of interest, i.e., the sample lies outside

of the measurement path.

Three papers have been published on our capacitance cell which attempted

to make the metrology as broadly applicable as possible. The first paper

described the use of the cell for CTE measurements of insulating materials.[10]

The second paper expanded the use of the capacitance cell to hygrothermal

expansion of insulating materials.[11] The third, and final, paper extended

the capability of the cell to measure insulating, semiconducting, and

conducting samples.[12] This recommended practice guide expands upon

the results presented in those three papers and gives specific construction

and assembly guidelines and discusses the format in which the resulting

data should be presented.

While in the broadest sense this recommended practice guide describes

measurement of ìhygrothermal expansion,î the main focus of this guide is the

determination of the coefficient of thermal expansion of thin film materials.
◆ Capacitance Cell Measurement

2

Since hygrothermal expansion is a multivariate function, the reference state

and the resultant relative expansion are not easily (or uniformly) defined.

Data can be examined that have been measured at:

● constant temperature and variable humidity;

● variable temperature and constant relative humidity;

● variable temperature and constant partial pressure of water vapor; or

● variable temperature and humidity.

No coefficient of hygrothermal expansion is easily defined. This difficulty

is compounded by the fact that relative humidity history is typically more

important than thermal history, as demonstrated in Ref. [11]. For these

reasons, while the necessary computation scheme will be presented for

determining the thickness under hygrothermal expansion, no discussion of

how the data should be presented will appear in this guide.
3

Scope ◆

2. Scope

This metrology is designed primarily for obtaining the z-axis expansion of

materials, with thicknesses between 2 μ m and 1 cm, such as those typically

used in High Density Interconnect Structures (HDIS) and microvias by the

use of a capacitance-based measurement technique. The expansion of these

materials can be due to changes in temperature (i.e., thermal expansion),

moisture (i.e., swelling), or any combination of these two (i.e., hygrothermal

expansion).
◆ Capacitance Cell Measurement

4

3. Test Specimens

3.1 Size

Specimens shall have thicknesses between 2 μm and 1 cm and have

in-plane dimensions of approximately 1 cm x 1 cm and be as close to

a perfect square as possible. Three such specimens are needed for

each test.

3.2 Condition

All specimens should be free from dust and as uniform as possible.

3.3 Preparation

All specimens should be fully cured according to manufacturerís

recommendations.
5

Apparatus ◆

4. Apparatus

4.1 Instrumentation

4.1.1 A precision automated three-terminal capacitance bridge which uses

an oven-stabilized quartz capacitor and has a cited guaranteed relative

resolution of better than 5 x 10ñ7 pF/pF for a range of capacitances of

0.625 pF to 1250 pF (equivalent to AndeenñHagerleen 2500A 1 kHz

Ultra-Precision Capacitance Bridge1).

4.1.2a Thermal Expansion Measurements

An oven capable of maintaining the temperature(s) of interest to

± 0.1°C with a feed-through hole for the coaxial cable and equipped

with a dry air (not nitrogen gas) purge (dew point of less than ñ70°C).2

4.1.2b Hygrothermal Expansion Measurements

An environmental chamber capable of maintaining the temperature(s)

of interest to ± 0.5°C and the relative humidities of interest to ± 0.5%.

4.1.3 Pressure sensor (or barometer) capable of measuring atmospheric

pressure to within 13 Pa (0.1 mm of Hg).

4.1.4 Vacuum pump and vacuum oven with an interior large enough for the

assembled capacitance cell.

4.1.5 Laminar flow hood equipped with a high efficiency particle arresting

(HEPA) filter, or ISO Class 6 cleanroom.

4.1.6 Analytical balance capable of determining the mass of the top

electrode to ± 1 g.

1 Certain commercial materials and equipment are identified in this guide in order to

specify adequately the experimental procedure. In no case does such identification

imply recommendation or endorsement by the National Institute of Standards and

Technology, nor does it imply that the items identified are necessarily the best

available for the purpose.

2 All cited uncertainties in this document are the best estimate of two standard

deviations in the experimental uncertainty.
◆ Capacitance Cell Measurement

6

4.1.7 A caliper capable of measuring an approximately 2 mm-thick bar to

± 0.1 mm.

4.1.8 (Optional) A computer capable of interfacing with the oven, the pressure

sensor, and the capacitance bridge to automate the measurements.

4.2 Capacitance Cell

4.2.1 Electrode Construction

The electrodes are constructed in the following manner: two 10 cmdiameter fused-quartz blanks are ground and polished to optical flatness.

On one side (the back) of one blank, which will be the bottom electrode,

a thin (0.1 mm) groove 1 cm in diameter and extending to within 5 mm

of the front surface is cut in the center of the blank. The circumference

of the bottom electrode has three small indentations drilled so that set

screws in the holder can be inserted into them without any applied force

(see Figure 1). In the center of both blanks, 2 mm diameter holes are

drilled through from the back surface to the front surface. On the

front side of both blanks, 3 mm diameter hemispherical indentations

are made into the surface of the blanks centered on the drilled holes.

(The dimensions for all of the above aspects of the electrode construction

are shown in Figure 1.)

Ball bearings (3 mm in diameter) are placed into the hemispherical

indentations on the front surface of the blanks and the blanks are

inverted. Into each of the holes that were capped with the bearing,

a piece of 4 cm long 16 gauge solid copper wire that touches the ball

bearing is inserted. By resting the wire on the ball bearing, the end

of the wire will be just short of the front surface of the electrode.

The wires are held centered in the hole and the holes are filled with a

high temperature (use temperature >200°C) conducting epoxy (volume

resistivity of less than 2 x 10ñ3 Ω cm). Emerson & Cuming Specialty

Polymers ECCOBOND SOLDER 58C has been found to be sufficient

for these purposes. After the epoxy has been B-staged (80% complete

cure), the bearing is removed and curing is completed.

◆ Capacitance Cell Measurement

6
7

Figure 1. Fused quartz blank electrode construction. Note that the drawing is not

to scale; the supplied measurements should be used. (The diameter of the blank is

10 cm as specified in the text.)

Apparatus ◆
◆ Capacitance Cell Measurement

8

Figure 2. Electrode construction showing the nichrome coating, placement of the

copper wires, conducting paint, and one of the three screw clamp holes. Note that

the drawing is not to scale; the supplied measurements should be used.
9

A nichrome coating (see Figure 2) is applied such that it covers all

surfaces except for a small area around the central wire on the back

side. The coating is removed on the front sides of both electrodes by

scribing on a 3 cm radius (width < 0.1 mm) on the top electrode, and on

a 1.5 cm radius on the bottom electrode, to form a guard gap of minimal

cross-sectional area. The scribing of these gaps should be performed in

such a manner so as to minimize the raising of material. In construction

of the electrodes, the main precautions are maintenance of flatness and

minimization of the guard gaps. To complete construction, the groove

(well) cut in the back of the bottom electrode is then filled with a

high-temperature conducting silver paint that touches the nichrome

on the outside edge but not the inner electrode, and the paint is cured

according to manufacturerís specifications.

4.2.2 Holder Construction

The holder, shown in cross-section in Figures 3, 4 and 5, has a land

machined in it, so the bottom electrode can rest on it without any applied

force. The drawings show all appropriate dimensions for construction

of the holder.

4.2.3 Cell Assembly

The bottom electrode is set in the holder. The safety screws in the

holder are aligned with the indentations in the circumference of the

bottom electrode. The screws are tightened until they are finger tight

and are then backed off by a o turn to eliminate strain on the electrode.

The holder and bottom electrode are then inverted. The shield plate,

shown in Figure 6, is then clamped in the bottom of the holder 1 cm

away from the bottom electrode with the copper wires passing through

the center of the holes in the guard plate. The screws along the radii

are finger-tightened to hold the shield plate in place.

The coaxial bridge low lead is made from poly(tetrafluoroethylene)

insulated coaxial cable attached at one end to a BNC receptacle in the

holder using high-temperature solder with a melting point above 250°C.

The other end is clamped to the shield plate placed between it and

the bottom of the electrode. Fine 30-gauge wire coil soldered, using

the high-temperature solder, to the poly(tetrafluoroethylene) insulated

cable is used to complete the contact to the electrodes to eliminate

any thermally-induced strains from the teflon insulation of the cable.

Apparatus ◆
◆ Capacitance Cell Measurement

10

Figure 3. Cross-sectional view of the dimensions for construction of the electrode

holder. The materials for the cell holder are brass or copper. Note that the drawing

is not to scale; the supplied measurements should be used.
11

Figure 4. Top and bottom views of the electrode holder. The materials for the cell

holder are brass or copper. Note that the drawing is not to scale; the supplied

measurements should be used.

Apparatus ◆
◆ Capacitance Cell Measurement

12

Figure 5. Side and bottom views of the dimensions for construction of the electrode

holderís bottom cover plate. Note that the drawing is not to scale; the supplied

measurements should be used.
13

Figure 6. Side and bottom views of the dimensions for construction of the aluminum

shield plate. Note that the drawing is not to scale; the supplied measurements

should be used.

Apparatus ◆
◆ Capacitance Cell Measurement

14

Actual connection is made by a push-on connection (which had the

30-gauge wire soldered to it using the high-temperature solder) using

center female contacts from 50 Ω BNC connectors. The center

conductor of the BNC receptacle connects with the center wire in

the electrode and the ìbraidî/outer connector with the outer wire in the

electrode. The above electrical connections are diagrammed in Figure 7.

The cover plate shown in Figure 5 is then placed on the bottom and is

fastened to the holder with pan head screws. The BNC receptacle is

connected to the capacitance bridge by low-noise, high-temperature

coaxial cable.

A miniature semirigid copper clad 50 Ω coaxial line (equivalent to

Pasternack Enterprises part number PE3823-18) clamped to the side

of the holder is used to make contact with the top electrode. The center

conductor connects to the center wire and the braid to the outer wire on

the electrode. This lead is connected by low-noise, high-temperature

coaxial cable to the high terminal of the capacitance bridge. This lead

overhangs the electrode and also makes contact through fine copper

coils. These coils do not need any shielding as the connection to the

bottom electrode is completely shielded from it by the holder. In this

way, all thermally induced strains from the connections are minimized

without compromising the necessary shielding. See Figure 8 for a

diagram of the completely-assembled cell.

4.3 Accessories

 The following or equivalent are recommended:

4.3.1 Coaxial Cable

Use low-noise, high-temperature insulated cables with high-quality

connectors equivalent to Kistler Corporation Model 1635C(X) coaxial

cable.
15

Apparatus ◆

Figure 7. Schematic of electrical connections to bottom electrode.
◆ Capacitance Cell Measurement

16

Figure 8. Diagram of assembled capacitance cell.
17

5. Procedure

The process of experimentally determining the CTE consists of two

procedures ó calibration and specimen tests. Calibration needs to be

performed only once after the sample cell has been constructed. For all

measurements, three specimens are need for each run. The specimens

should be as free from dust and surface contamination as possible. Cell

assembly should be performed in the laminar flow hood or cleanroom.

The electrode surfaces are cleaned with distilled water followed by

ethanol and dried with lens tissue. High pressure filtered air is used

to remove any remaining particles from the surface. All samples are

cleaned in the same way unless they are soluble or swell in the presence

of ethanol or water. The samples are placed on the bottom electrode

at the outer edge of the outside guard ring and are spaced 2π/3 radians

(120°) apart. The entire cell apparatus is then placed in the vacuum oven

at 25°C at a pressure of less than 1 kPa for 1 h. (Evacuation of the cell is

necessary to simulate wringing, i.e., the removal of the trapped air layer

caused by the contact of optically smooth surfaces.)

5.0a Thermal Expansion

The assembled cell is transferred to an oven equipped with a dry air

purge. The cell is equilibrated 10°C above the highest temperature of

interest for approximately 1 h. Measurements are performed after the

cell assembly has dimensionally equilibrated. Dimensional equilibration

is determined by the calculated thickness (defined below) changing by a

relative value of less than 1 x 10ñ6 over a period of 1 h with measurements

being performed every 10 min or less (the process can be automated with

all measurements performed by a computer). It should be noted that

dimensional equilibration will occur either with, or after, thermal equilibration

ó the latter occurring when the material undergoes volume recovery

or some other relaxation process.

The thickness of the sample (d ) is given by the following equation (with

all constants given in SI units):

( )

( ) 



 





−

ε + = Pp RTPp RT

C

A d

1 /

0 2 / 1

(1)

where:

ε0 = 8.854 pF/m is the permittivity of free space

R = 8.314507 L kPa molñ1 Kñ1 is the gas constant

Procedure ◆
◆ Capacitance Cell Measurement

18

P = 4.31601 x 10ñ3 L/mol is the molar polarization of dry air[13]

C = the measured capacitance

p = the measured atmospheric pressure

T = the cell temperature

A = the effective electrode area, which is a function of temperature and

must be calibrated

The temperature is then changed and the cell is re-equilibrated.

5.0b Hygrothermal Expansion

The cell is transferred to an environmental chamber that has been

pre-equilibrated to the initial temperature (T) and relative humidity (HR)

of interest. The thickness of the sample is given by[14]:

 (2)

where ε is the dielectric constant of the air in the gap between the

electrodes, determined by the sample thickness. The determination of

ε is given in Appendix A. It is easily shown that under dry conditions,

ε reduces to the form in brackets in Eq. (1). T and/or HR are then

changed and the cell is re-equilibrated.

5.1 Calibration

Calibration of the cell involves determination of the effective electrode

area A as a function of temperature T. The specimens used for the

calibration must have a CTE (α) of less than 0.05 x 10ñ6 m/m and be

dimensionally stable (i.e., do not significantly deform under loading).

Zerodur has been found satisfactory for this purpose.[15] Flat samples

with a thickness of approximately 2 mm should be used. The specimens

should be measured with the caliper at room temperature, and room

temperature should be measured to ± 0.1°C. The value for the

approximate room temperature thickness is determined by averaging the

three sample thicknesses. While the uncertainty with this calculation will

limit the absolute thickness values, it will not affect the uncertainty in the

expansion values.

ε

ε = CA d 0 ε

ε
19

Procedure ◆

The procedure in Section 5.0 is followed and measurements are

performed over the entire temperature range of interest with a minimum

of five different temperature measurements. The rearranged form for

Eq. (1) that is used for calibration of A is:

 (3)

where:

drt = average calibration specimen thickness at room temperature

Trt = room temperature

α = CTE of the calibration specimen material

From the measured area (A) as a function of temperature, the radius (r)

is determined from:

(4)

The (r , T) data is then fit by a linear least squares process, which is

available in most spreadsheets, to a quadratic equation of the form:

(5)

(It should be recalled that ìlinearî in the linear least squares refers to

linearity in the coefficients not a linear equation.) Equation (5) is used to

determine the area in Eqs. (1) and (2) as a function of temperature used

in all subsequent sample measurements.

The calibration run should also be used as a verification of expected

reproducibility. The reproducibility of thickness on temperature cycling

should be on the order of 2 x 10ñ6 μm/μm under dry atmosphere

conditions. If the reproducibility is an order of magnitude greater than this,

either there are problems with the connections in the cell, there is too much

vibration in the environmental chamber, or there is a problem with the

userís technique. See Note 6.4 for suggested ìdebuggingî techniques.

( )

[ ] ( ) [ ] ( ) drt drt T Trt Pp RT

C Pp RT A + α − 



 





+

−

ε = 2 / 1

1 /

0

r = A/ π

2 3

2

r = A1T + A T + A
◆ Capacitance Cell Measurement

20

5.2 Test Measurements

The x-y dimensions of each of the samples are measured (to determine

the total surface area), and the mass of the top electrode is determined

using the analytical balance. The nominal loading (Force/Area) is then

determined by the following equation:

(6)

Where:

m = Mass of the top electrode

g = 9.80 m/s2, the gravitational acceleration constant

x1,y1 = x and y dimensions of sample 1, respectively

x2,y2 = x and y dimensions of sample 2, respectively

x3,y3 = x and y dimensions of sample 3, respectively

The capacitance cell is loaded with the three specimens and measured

at a minimum of five temperatures in the manner described in Section 5.0.

The temperatures must be measured in a fixed step decreasing order.

5.3 Evaluation

It must be determined whether or not dimensional changes have occurred

during the sample run. If the thickness (d) versus time (t) plot appears like

Figure 9 rather than Figure 10, then creep effects are present. The former

curve is a convolution of the thermal equilibration process and viscoelastic

or plastic deformation of the sample. An estimation of the thermal

equilibration time can be made by using samples of known thermal

conductivity and diffusivity that are dimensionally stable and using the

appropriate heat transfer equations as could be found for example in

references 16 and 17. Viscoelastic or plastic deformations are more

difficult to correct for ó one approximate method is given in Ref. [10].

1 1 2 2 3 3 x y x y x y

mg

Area

Force Loading

+ + = =
21

Figure 9. Typical thickness versus time plot for a material that undergoes

dimensional contraction (creep) under load after a temperature step. (This plot is

given to aid the reader only; it should not be interpreted as actual data.)

Figure 10. Typical thickness versus time plot for a material that does not undergo

dimensional contraction (creep) under load after a temperature step. (This plot is

given to aid the reader only; it should not be interpreted as actual data.)

Procedure ◆
◆ Capacitance Cell Measurement

22

ref

T ref

T d

d d

E − =

3 4

2

2

3 ET = B1T + B T + B T + B

2 3

2 3B1T 2B T B

dT

dE

α = = + +

5.4 Calculation of Coefficient of Thermal Expansion

To determine the CTE, the relative expansion is calculated using a

reference temperature near room temperature (preferably 20°C or

25°C). The relative expansion data are then given by:

 (7)

where:

ET = the relative expansion at temperature T

dref = the thickness at the reference temperature

The expansion data are then fit by a linear least squares procedure to a

cubic polynomial of the form:

 (8)

The CTE is then given by:

(9)

5.5 Report of CTE Data

5.5.1 Report Eq. (9) along with the standard deviation in the coefficients.

5.5.2 Report the mean value for the CTE calculated by Eq. (9) using a

temperature halfway between the highest and lowest measured

temperatures.

5.5.3 Report the reference temperature.
23

5.5.4 Report the temperatures that were measured.

5.5.5 Report the applied loading [Eq. (6)].

5.6 Plot of CTE Data

5.6.1 Plot the relative expansion (μm/m) versus temperature (°C) for the

specimen, such as is shown in Figure 11. Include a curve generated

from the cubic fit [Eq. (8)].

Figure 11. Typical relative expansion curve. (This plot is given to aid the reader

only; it should not be interpreted as actual data.)

Procedure ◆
◆ Capacitance Cell Measurement

24

6. Notes

6.1 Examine the samples after the test. If the sample shows signs of

excessive load or distortion, a repeat run should be performed in which

the length of the samples are increased by a factor of two which will

result in a decreased pressure.

6.2 Better estimates of the reproducibility of expansion data can be obtained

by performing multiple temperature cycles in the calibration procedure.

The difference between subsequent measurements at the same

temperature shows the degree of reproducibility achievable with the

material under test.

6.3 The dissipation factor (D = tan δ) measured by the capacitance bridge

should be less than 1 x 10-4 rad if all of the electrical contacts have been

made properly.

6.4 ìDebuggingî a Reproducibility Problem

The first thing to check is cell assembly. The resistances of the

connections should be measured using a digital voltmeter. The bottom

electrode should be checked while connected to the holder. One lead

should be connected to the center conductor of the BNC receptacle in

the holder. A piece of soft lead-based solder is then curved, and the

curve is touched to the surface of the nichrome inner electrode while

the other voltmeter lead is clamped to the piece of solder. (Repeat for

the outer conductor of the BNC receptacle and the nichrome outer

electrode.) If the resistance is higher than approximately 50 Ω, there is

a problem with the electrical connection. This should be traced down by

disassembling the cell and checking each connection: BNC receptacle to

30 gauge coil and copper lead from the back side of the bottom electrode

to the nichrome surface. A similar check is performed with the top

electrode. If all electrical connections are within the above-specified

tolerance, then vibration must be examined as a problem. A metal box

can be employed so that the airflow over the cell is reduced to eliminate

some vibration. A sheet of silicone rubber can be used between ñ50°C

and 100°C underneath the cell to minimize vibration. If this does not

eliminate the reproducibility problem, then the user should examine their

technique for assembling the cell and performing the measurements to

try to find the reason behind the uncertainty.

A check should also be made to ensure that the 30 gauge wires have

remained soft coils and cannot exert any significant strain on the electrodes.

If they stretched out, this may introduce a reproducibility problem.
25

APPENDIX A ó DIELECTRIC CONSTANT OF AIR

To determine the dielectric constant of air containing water vapor, we first

need to determine the molar volumes of the water vapor and air. Dry air can

be treated as an ideal gas over the range of temperatures from ñ50°C to 150°C.

The molar volume of dry air is given by:

(10)

where pair is the partial pressure of air.

(11)

yields the saturation pressure of water in units of Pascal, where

F0 = ñ6043.6117 K, F1 = 18.9318833, F2 = ñ2.8238594 x 10ñ2 Kñ1,

F3 = 1.7241129 x 10ñ5 Kñ2, and F4 = 2.858487, at temperature Tchamber ñ the

temperature of the environmental chamber in Kelvin.[18] (Note that where no

subscript is present on T, it refers to the temperature of the capacitance cell.)

The partial pressure of water (pw) is then calculated by:

(12)

where HR is the relative humidity with values ranging from 0 to 1. The molar

polarization of water is temperature-dependent and is given in the literature to

be[13]:

(13)

The partial pressure of air is:

(14)

where patm is the barometric (atmospheric) pressure. The compressibility

factor for saturated water vapor (z = pv/RT) can be fit to a cubic equation

between 0°C and 150°C, i.e.,

air

air

p

RT

v =

( )



 



 = + + chamber + chamber + chamber

chamber

F F T F T F T

T

F

p ln 4

2

1 2 3

0

sat

pw = psat HR

-1

1

w 0.0039 L mol 20.847 K L mol = +

−

T

P

pair = patm − pw

Appendix A ◆
◆ Capacitance Cell Measurement

26

AT BT CT D

RT

pv z = = + + + 3 2

( )HR z′ = 1− 1− z





 





′ = ×

z RT

p P

v

P w

w

w

w

w w air air

air

air

w

w

total P P

v

P v

v

P v P = χ + χ 















+













 =

(15)

where A = ñ1.026 x 10ñ8 °Cñ3, B = ñ1.289 x 10ñ7 °Cñ2, C = ñ3.382 x 10ñ5 °Cñ1,

and D = 0.9998.[11,19] The compressibility factor increases towards unity at

lower relative humidities, therefore the relative humidity corrected value of the

compressibility factor must be utilized. The corrected value (z′ ) is[20]:

(16)

The molar polarizability per unit volume is then given by:

(17)

We assume additivity for the molar polarizabilities, i.e.,

(18)

where χi is the mole fraction of the ith component. Rearranging Eq. (18) to

obtain Ptotal/v yields:

(19)

and from this, the dielectric constant of the medium can be calculated through

the definition:

(20)

















+













 =

air

air

w

total w

v

P

v

P

v

P





 



 −

 +



 





ε =

v

P

v

P

total

total

1

1 2
27

APPENDIX B ó DISCUSSION OF UNCERTAINTIES

B.1. Electrode Tilt

An advantage of this cell is that any uncertainties due to the small amount

of tilt caused by sample variations will not significantly affect the results.

By simple symmetry arguments, it can be shown that any deviations due

to a lack of parallelism enter as the square of the tilt angle. For thin films

placed between 5 cm diameter plates, the maximum possible angle is so

small, approximately 1 μm/5 cm, that any variation in this angle is well

beyond any possible experimental resolution.

B.2. Temperature and Pressure Uncertainties

Under dry atmosphere conditions, the effect of an uncertainty in

measured temperature and pressure can be easily determined. For

a 1°C standard uncertainty in temperature, the combined standard

uncertainty in thickness is obtained by differentiating Eq. (1) with

respect to temperature, i.e.,

(21)

For a 1°C standard uncertainty at 25°C and 101.325 kPa, for a 100 mm-thick

sample, the relative expanded uncertainty in thickness is approximately

2 x 10ñ6. Better control of the temperature (as required by this guide)

reduces this uncertainty by an order of magnitude.

A similar equation can be derived for the combined standard uncertainty

in pressure, i.e.,

(22)

For a 0.13 kPa (1 torr) standard uncertainty in pressure, the relative

combined standard uncertainty is approximately 5 x 10ñ7 under the same

conditions given above.

The uncertainties caused by moisture are an order of magnitude greater.

In Ref. [11], we plotted the combined standard uncertainty dε/dHR

at 60°C as a function of HR. At 80% relative humidity, the relative

combined standard uncertainty in the thickness is approximately 25 μm/m.

( )2

0 3 ( ) d

d

C Pp RT

APpR d

T −

− ε =

( )2

0 3 ( ) d

d

C Pp RT

APRT d

p −

ε =

Appendix B ◆
◆ Capacitance Cell Measurement

28

B.3. Thickness Dependence of the Effective Area

The effective area A was determined for a thickness of approximately

2 mm. This effective area decreases as the film thickness decreases due

to a decrease in the fringing fields. The amount of the decrease can be

calculated under certain limiting cases. For the case examined here

where the dielectric constants of the substance between the electrodes

(air) and the substance in the guard gap (air) are equal and where we

assume a thick electrode, the thickness correction factor is[21,22]:





 



 δ ξ = + − t t

R

R

t

d

2 1 (23)

where:















 



 



 +

π

 −



 





π = δ −

2

1

2

ln 1 2

2

tan

2 2

d

t

t

d

d

t

t (24)

and where R is the electrode radius, d is the sample thickness, and t is

the guard gap width. The measured sample thickness [derived from

Eq. (1)] is multiplied by the correction factor corresponding to that

sample thickness and is divided by the correction factor corresponding

to the calibration sample thickness described in Section 5.1.

This correction is necessary only for reducing the error in the absolute

thickness, since it is minimized in the relative expansion computation

[Eq. (7)]. However, it should be realized that there is some uncertainty

introduced into the relative expansion due to the change in the effective

area induced by thermal expansion. To determine the magnitude of this

combined standard uncertainty, d in the above equation is replaced by:

d = d0 + αd0T (25)

where, for ease of computation, we have set the reference state as 0 K.

If we then differentiate with respect to temperature, we obtain:

( ) 















+ α

+

π

α = ξ

2

0 0

2

0

2 2

ln 1 2

d

d

d Td

g

R

d

T

d

(26)
29

Appendix B ◆

To examine the maximum combined standard uncertainties associated

with this, we assume a CTE (α) of 200 x 10ñ6 Kñ1. We then set

g = 0.1 mm, R = 1.5 cm, and T = 200 K (the maximum temperature

range for this system, i.e., from ñ50°C to 150°C) and solve for d0

corresponding to the maximum combined standard uncertainty.

The maximum combined standard uncertainty occurs at a thickness

of approximately 25 μm. The relative combined standard uncertainty

is approximately 67 x 10ñ6 over the entire temperature range (or

0.335 x 10ñ6/ °C). This is then compared to the total relative expansion

of the sample αT = 4 x 10ñ2, which indicates that it would introduce

a combined standard uncertainty in the calculated CTE of, at most,

0.2% under the worst case possible. This level of uncertainty in the

CTE is negligible, and, hence, these contributions to the combined

standard uncertainty can be neglected.

B.4. Other Sources of Uncertainty with Respect to

 Thickness Reproducibility

One source of uncertainty that is introduced from three pieces that

are not of identical height arises if the sample pieces move during the

course of the measurement. If the samples are not in the exact same

location and orientation, the mean distance between the electrodes

will change simply due to the geometry. This source of uncertainty is

minimized by eliminating all sample motion relative to the electrodes.

Furthermore, excess vibration should be eliminated as this can introduce

a time-dependent thickness variation. In the case of optically smooth

samples, the wringing process described previously will minimize sample

sliding just due to sample friction.

Our best estimate for the total magnitude of the wrung layer was derived

from measurements on <0001> single crystal Al2O3. This sample was

optically flat and sufficiently rigid that any additional sources of uncertainty

were eliminated. From data taken on the sample thickness after initial

assembly of the capacitance cell to the thickness after wringing (i.e.,

after vacuum evacuation ó described in Section 5.0), an estimated

thickness of greater than 40 nm was determined for the combined top

and bottom interfaces. However, proper sample handling and

conditioning have been shown to eliminate this source of error.

Another source of uncertainty is that due to creep. For many materials,

such as filled epoxy resins and polyimides, there is no observable creep

near 25°C and low humidities. However, especially at high temperatures

and humidities, the sample can creep, as shown by a downward drift in

the total distance after thermal equilibration is achieved. In addition,
◆ Capacitance Cell Measurement

30

there can be dimensional relaxation as well as diffusion-controlled

moisture absorption, all on longer time scales than that due to thermal

equilibration. Ultimately, these dimensional changes will set the limit as

to how well thermal expansion can be defined. Since this is a material

based limit, this uncertainty must be estimated for each sample

individually and is not due to the measurement cell.

Similarly, any warpage can interfere with the actual measurement.

For composite samples with layers of different CTEís in the plane of

the sample, warpage can become a problem. This is one case where

increased load could help. However, we have observed a highly-stressed

sample that formed visible wrinkles over a narrow temperature range.

This was manifested as a sudden jump in sample thickness.

Finally, sample flatness becomes an increasing problem when dealing

with anisotropic materials. For some liquid crystal polyimides, the thermal

expansion can be close to zero in the xy plane while near 100 x 10ñ6 Kñ1

in the thickness direction. For these materials, sample flatness will

become the limit to the estimation of the CTE perpendicular to the xy

plane. As an extreme case, the largest uncertainties due to anisotropy

will be if any wrinkles exist (especially if the xy expansion is small and

the z axis expansion is large) since there will be a convolution of the xy

and z axis CTEís in the measured CTE. For materials that are not highly

anisotropic, this source of uncertainty becomes less important, and in the

limit of an isotropic material, the orientation and shape of the sample do

not matter.

B.5. Capacitance Bridge Calibration

The calibration of the capacitance bridge for absolute capacitance is

not an essential component of these measurements. If the capacitance

bridge is stable and linear (as is normally the case with capacitance

bridges with oven-stabilized fused quartz references), then once

the calibration has been performed, only the relative values of the

capacitance will matter. Errors caused by miscalibration of absolute

capacitance, or by not correcting for cable length, resistance, etc.,

will be eliminated in this manner.
31

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References ◆
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32

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33
◆ Capacitance Cell Measurement

34

November 2001