Stresses in Cu Thin Films and Ag/Ni Multilayers - Harvard School of ...

Stresses in Cu Thin Films and Ag/Ni Multilayers 

A thesis presented 

by 

Dillon Dodd Fong 

to 

The Division of Engineering & Applied Sciences 

in partial fulfillment of the requirements 

for the degree of 

Doctor of Philosophy 

in the subject of 

Applied Physics 

Harvard University 

Cambridge, Massachusetts 

September 2001

c○2001 - Dillon Dodd Fong 

All rights reserved.

Thesis advisor Author 

Frans Spaepen Dillon Dodd Fong 

Stresses in Cu Thin Films and Ag/Ni Multilayers 

Abstract 

The stresses in substrate-bonded Cu thin films and Ag/Ni multilayers were studied using 

wafer curvature and x-ray scattering techniques. The biaxial modulus of sputtered Cu films 

in the as-deposited state is less than that of the dynamic by approximately 15%. After 

annealing, the modulus increases due to enhanced 〈111〉 texture but still remains compar- 

atively low. The deficit increases at higher temperatures, suggesting anelastic processes 

contribute to the modulus reduction. The annealed Cu films show high flow strengths due 

to the presence of the rigid substrate. Attempts were made to find a dislocation boundary 

layer in the plastically strained film by grazing incidence diffraction studies. No confirma- 

tion of a strain gradient near the substrate could be found, however, due to the insensitivity 

of the measurement to strains at buried interfaces. Significant strain relaxation was seen at 

the surface, however. The effect of interphase boundaries on internal stresses was also stud- 

ied in Ag/Ni multilayers. The {111} interface stress was found to be −2.17 ± 0.15Nm −1 , 

in agreement with previous studies. High and low-angle x-ray scattering experiments show 

that this value is unaffected by intermixing. The widths of Ag/Ni and Ni/Ag interfaces were 

determined and correlated to an excess volume that derives from the volume expansion of 

misfit dislocations. A positive excess volume is consistent with a negative interface stress 

when accounting for anharmonic terms in the interatomic potential.

Contents 

Title Page ...................................... i 

Abstract ....................................... iii 

Table of Contents .................................. iv 

List of Figures ....................................viii 

List of Tables ....................................xiii 

Acknowledgments .................................. xiv 

Dedication ......................................xvii 

1 Background 1 

1.1 Elasticity ................................... 2 

1.2 Dislocation Plasticity ............................. 6 

1.3 Thin Film Growth and Microstructure . ................... 10 

1.4 Two Common Stress Determination Techniques ............... 10 

1.4.1 Wafer Curvature ........................... 11 

1.4.2 X-Ray Diffraction Measurements .................. 14 

1.5 Sputter Deposition .............................. 16 

2 The Elastic Properties of Cu Thin Films 19 

2.1 The Variation of Stress with Temperature in a Thin Film on an Elastic Substrate 

...................................... 21 

2.2 The Biaxial Modulus Measurement . . . ................... 23 

2.2.1 The Effect of Texture on the Biaxial Modulus ............ 24 

2.3 Experiment .................................. 25 

2.3.1 Film Preparation ........................... 25 

2.3.2 Film Characterization ......................... 26 

2.3.3 X-Ray Diffraction .......................... 26 

2.3.4 Curvature Measurements ....................... 27 

2.4 Results ..................................... 28 

2.4.1 Microstructure ............................ 28 

2.4.2 Film Texture ............................. 28 

2.4.3 Biaxial Modulus ........................... 36 

iv

Contents v 

Measuring Both the Biaxial Modulus and the CTE . ........ 36 

As-deposited and Annealed Films at Room Temperature ...... 37 

Biaxial Modulus as a Function of Temperature . . . ........ 39 

Biaxial Modulus as a Function of Oscillation Frequency ...... 40 

2.5 Discussion ................................... 43 

2.5.1 Texture . . .............................. 43 

2.5.2 Cause of the Modulus Deficit .................... 44 

2.6 Conclusions and Future Research ....................... 47 

3 The Plastic Properties of Cu Thin Films 49 

3.1 The Current Understanding of Plastic Properties of Thin Films ....... 49 

3.1.1 High Temperature Deformation ................... 50 

3.1.2 Low Temperature Deformation . ................... 51 

Discrete Dislocation Models . . ................... 54 

Strain Gradient Plasticity in Thin Films ............... 59 

3.2 Determining the Strain Profile ........................ 63 

3.3 Experiment .................................. 64 





3.4 Results ..................................... 68 

3.4.1 Microstructure ............................ 68 

3.4.2 Curvature Results ........................... 70 

3.4.3 X-Ray Diffraction Results . . . ................... 72 

Measurements from the Surface of the Film ............. 72 

Measurements from the Film/Substrate Interface . . ........ 73 

Comparison with sin2ψ 2ψ 2ψ Measurements ............... 74 

3.5 Discussion ................................... 75 

3.5.1 The As-Deposited Films ....................... 77 

The Strain Profile........................... 78 

Anisotropy .............................. 78 

3.5.2 The Thermally Cycled Films . . ................... 79 

The Strain Profile........................... 79 

Anisotropy .............................. 80 

3.6 Conclusions and Future Research ....................... 81 

4 The Ag/Ni {111} Interface Stress 82 

4.1 The Ag/Ni {111} Interface .......................... 82 

4.1.1 The Structure and Energy of Semicoherent Interfaces ........ 83 

Structure of the Ag/Ni {111} Interface ................ 83 

Energy of the Ag/Ni {111} Interface ................. 88

Contents vi 

4.2 Interfacial Thermodynamics ......................... 89 

4.2.1 Fluid-Fluid Interfaces ......................... 89 

4.2.2 Solid-Fluid Interfaces ......................... 90 

4.2.3 Solid-Solid Interfaces ......................... 92 

4.3 Principle of the Interface Stress Measurement ................ 95 

4.4 Previous Measurements of the Ag/Ni {111} Interface Energy and Stress . . 96 

4.5 Experiment .................................. 98 





4.6 Results .....................................100 

4.6.1 Microstructure ............................100 

4.6.2 Substrate Curvature Stress . . . ...................101 

4.6.3 Volume Stress .............................101 

4.6.4 The Interface Stress ..........................105 

4.7 Models for the Compressive Interface Stress .................107 

4.7.1 Intermixing ..............................107 

A Systematic Method of Extracting Strain and Composition Profiles 

of Multilayers . . ...................110 

Results from Diffuse Scattering ...................113 

4.7.2 Excess Volume / Misfit Dislocations .................117 

Excess Volume ............................122 

Misfit Dislocations ..........................123 

4.8 Conclusions and Future Research .......................125 

Bibliography 126 

A X-Ray Stress Analysis 145 

A.1 Untextured Thin Films ............................147 

A.1.1 Biaxial Stress Case ..........................148 

Equibiaxial Stress Case ........................148 

A.2 Textured Thin Films in Equibiaxial Stress ..................149 

A.3 The Strain-Free Lattice Parameter in Biaxially Stressed Samples ......150 

B X-Ray Errors 152 

B.1 Mandatory Corrections ............................153 

B.1.1 Lorenz-Polarization-Absorption Corrections to the Intensity ....153 

B.1.2 Refraction Corrections ........................154 

B.2 Geometric Errors . ..............................155 

B.2.1 Receiving Position Errors . . . ...................155 

B.2.2 Horizontal Divergence Errors . ...................157

Contents vii 

B.2.3 Alignment Errors ...........................157 

Specimen and Beam Displacement ..................157 

Effect of the ω-Axis Not Corresponding to the 2θ-Axis .......159 

B.3 Corrections for a Position Sensitive Detector .................159 

B.3.1 Parallax Corrections for a PSD . ...................159 

B.3.2 Small Beam Spot Corrections . ...................160 

B.3.3 PSD Calibration Constant Errors ...................160 

B.4 Counting and Fitting Errors ..........................160 

C Alignment Procedure for the Harvard Four-Circle Diffractometer 161 

C.1 Coarse Alignment . ..............................161 

C.2 Fine Alignment . . ..............................162 

C.3 Aligning Samples . ..............................163 

C.4 Calibrating the Position Sensitive Detector ..................164 

D Elastic Constants 165 

D.1 Important Elastic Quantities .........................165 

D.2 The Rotation Matrix .............................166 

D.3 Average Elastic Quantities For Polycrystalline Aggregates . ........167 

D.4 The Vook-Witt Model of Grain Interaction ..................167 

D.4.1 The Hornstra-Bartels Calculation Technique .............169 

D.4.2 Results with the Vook-Witt Model ..................171 

E The Stoney Equation 175 

F List of Symbols 179

List of Figures 

1.1 Interatomic potentials (a) and their first (b) and second (c) derivatives. 

Both the Lennard-Jones potential (solid) [140] and a harmonic potential 

(dashed) are shown. .............................. 4 

1.2 (a) A depiction of the vertical force balance between dislocation line tension 

and the Peach-Koehler force. (b) An illustration of a dislocation bowing 

between two impenetrable obstacles with separation ℓ. ......... 8 

1.3 A schematic showing how the film stress causes substrate curvature. (a) 

The film and substrate are both shown in their strain-free state. (b) The 

film is stretched to fit onto the substrate. (c) The film is “glued” onto the 

substrate. (d) The film/substrate system bends in order to achieve force 

and moment balance. ............................. 12 

1.4 A schematic of the curvature measurement apparatus at Harvard University. 

Adapted from Mullin [158]. ......................... 13 

1.5 A schematic of the four-circle Huber diffractometer at Harvard University. . 16 

1.6 A schematic of different scattering geometries. (a) The symmetric θ − 

2θ geometry and χ rotation axis, (b) the geometry used for longitudinal 

diffuse scattering, and (c) the grazing incidence diffraction geometry. . . . 17 

1.7 A schematic of the sputtering chamber at Harvard University. ........ 18 

2.1 Stress as a function of temperature for a 1.91 µm sputtered Cu film on 

Si(100). The dotted lines show the hypothetical elastic stress in the film 

if no relaxation could take place (thermoelastic lines), and the labels (A), 

(B), and (C) denote elastic thermal cycles as described in the text. The 

upper thermoelastic line was drawn using the elastic constant data found in 

Bechmann et al. [12] and assuming mild 〈111〉 texture. . . . ........ 21 

2.2 Plan-view TEM micrographs of 0.87 µm thick copper in the as-deposited 

state (a) and after annealing at 600 ◦ C for 0.5 hrs (b). ............ 29 

2.3 A cross-sectional TEM micrograph of 1.90 µm thick copper grown on 

Ge(111). .................................... 30 

viii

List of Figures ix 

2.4 (a) Powder diffraction x-ray spectra for sputter-deposited copper films on 

Ge(111) (dashed), α-Al2O3(0001) (dotted), and Si(100) (solid). (b) (111) 

fiber texture plot of copper films on Ge(111) (dashed) and Si(100) (solid). . 31 

2.5 (a) Powder diffraction spectra, and (b) a (111) fiber texture plot for asdeposited 

and annealed copper films on Si(100). .............. 32 

2.6 Powder diffraction spectra for annealed copper films on Si(100) (solid) and 

α-Al2O3(0001) (dotted). ........................... 33 

2.7 (111) fiber texture plot of as-deposited and annealed 0.50 µm thick copper 

films. ..................................... 33 

2.8 (111) fiber texture plot of an electron beam evaporated copper film. .... 34 

2.9 The calculated biaxial modulus for 〈111〉 texture (a) and 〈100〉 texture (b) 

as a function of the texture pole standard deviation. A Gaussian texture 

pole is assumed. . . .............................. 35 

2.10 Plot showing how the biaxial modulus and CTE can be determined for a set 

of different substrates. ............................. 37 

2.11 Biaxial modulus as a function of the maximum annealing temperature exposed 

to a 1.88 µm thick copper film on Si(100). .............. 38 

2.12 Biaxial modulus as a function of the number of times thermally cycled to 

600 ◦ Cfora1.91 µm and 1.93 µm thick copper film. ............ 39 

2.13 Stress as a function of temperature for a 1.19 µm thick Cu film on Si(100). 

The elastic thermal cycles for the measurements of biaxial modulus at five 

different temperatures are shown for the as-deposited film (a) and the thermally 

cycled film (b). ............................. 40 

2.14 Biaxial modulus as a function of temperature for as-deposited copper films. 

The error bar denotes two times the standard deviation. The shaded area 

represents the calculated biaxial moduli after accounting for texture. It is 

based on a value of 226 ± 22 GPa at room temperature (cf. Table 2.4) and 

has the temperature dependence found in Simmons and Wang [207]. .... 41 

2.15 Biaxial modulus as a function of temperature for a copper film previously 

annealed at 600 ◦ C. The error bar denotes two times the standard deviation. 

The shaded area represents the calculated biaxial moduli after accounting 

for texture. It is based on a value of 256 ± 22 GPa at room temperature 

(cf. Table 2.4) and has the temperature dependence found in Simmons and 

Wang [207]. .................................. 42 

2.16 Biaxial modulus as a function of temperature for copper films previously 

thermally cycled to 600 ◦ C for different strain oscillation frequencies. . . . 43 

2.17 Depiction of dislocation bowing at high temperatures. Thermal fluctuations 

help the dislocation to bow past weak obstacles, but the radius of curvature 

remains below that required for slip. . . ................... 46

List of Figures x 

3.1 Diagram showing the nucleation and propagation of dislocations along a 

slip plane from the surface to the grain boundaries and film/substrate interface. 

...................................... 55 

3.2 Schematic showing the predicted dislocation behavior at different points 

on the stress-temperature profile. Based on the diagram by Weihnacht and 

Brückner [245]. . . .............................. 57 

3.3 The scattering geometry for the GID measurements. As shown in (a), the 

2θ value is not the same as the motor position for the detector, 2θ ′ . The correction 

can be made using the well known formulas for the right-spherical 

triangle shown in (b). ............................. 67 

3.4 Cross-sectional views of an as-deposited (a) and annealed (b) 0.87 µm 

thick copper film................................ 69 

3.5 Deposition stress for ion beam sputtered Cu films as a function of total film 

thickness. ................................... 70 

3.6 Room temperature flow strength as a function of inverse thickness for Cu 

samples thermally cycled to 660 ◦ C. ..................... 71 

3.7 The stress-temperature profiles for the Cu films used in the strain gradient 

studies. .................................... 71 

3.8 Lattice parameter of CeO2 as a function of incidence angle, α. The lattice 

parameters were measured with the (331) reflection. ............. 72 

3.9 Elastic strain gradients in Cu films. The solid and dotted curves refer to the 

thermally cycled films and as-deposited films, respectively. The reflection 

is denoted by different markers, as shown in the legend. The plastic strain 

is given by 0.9% minus the elastic strain. ................... 73 

3.10 sin 2 ψ-plot for an as-deposited Cu film. ................... 74 

3.11 sin 2 ψ-plot for a thermally cycled Cu film................... 75 

3.12 Hypothetical strain profiles and their normalized finite thickness Laplace 

transforms for a 1.8 µm thick Cu film. .................... 77 

4.1 The Ag/Ni phase diagram [149]. ....................... 85 

4.2 Top view of the Ag/Ni {111} interface, with Ag and Ni denoted by closed 

and open circles, respectively. After relaxation, misfit dislocations are 

formed, as delineated by the dark lines. The coherent areas and intrinsic 

stacking fault areas lie in the centers of the white and shaded triangles, 

respectively. .................................. 86 

4.3 An edge-on view of the Ag/Ni {111} interface along the [110] direction, 

with the Ag and Ni atoms denoted by closed and open circles, respectively. 

The large circles are atoms in the plane of the diagram; the small circles are 

atoms in the planes immediately above and below the plane of the figure. 

Two distinct dislocation nodes can be seen and associated with the different 

stacking conditions. Adapted from Gao and Merkle [75]. . . ........ 87

List of Figures xi 

4.4 A schematic demonstrating how misfitting surface layers can lead to internal 

bulk stresses. . .............................. 93 

4.5 A schematic showing how interface stresses can lead to internal bulk stresses. 93 

4.6 A cross-sectional view of a Ag/Ni multilayer with � = 4.2nm........101 

4.7 Offset low-angle θ − 2θ diffraction spectra for the three smallest bilayer 

thicknesses. The spectra were taken with an x-ray energy of 8020 eV. . . . 102 

4.8 (a) Volume stresses in Ni (closed squares) and Ag (open squares) layers as 

a function of inverse bilayer thickness. (b) Substrate curvature stress (open 

circles) and average volume stress (closed circles) as a function of inverse 

bilayer thickness. The error bars for all the data are within the size of the 

symbols. The arrow drawn from (a) to (b) illustrates how the layer stresses 

are averaged to give 〈σ 〉x−ray. .........................103 

4.9 Offset GID spectra for Ag/Ni multilayers having the indicated bilayer thickness 

�. The vertical lines indicate the {220} peak positions for strain-free 

bulk Ag and Ni. The x-ray energy was 8020 eV. ...............104 

4.10 In-plane strains in nickel and silver as a function of bilayer thickness. . . . 105 

4.11 Difference between the substrate curvature stress and the average volume 

stress for Ag/Ni multilayers as a function of inverse bilayer thickness. The 

line is a weighted least-squares fit for the three smallest bilayer thicknesses. 106 

4.12 The average {111} d-spacing for nickel. Since the nickel is in tension parallel 

to the {111} planes, d111 is expected to be less than the strain-free 

d-spacing of 0.20345 nm for a positive Poisson ratio. ............108 

4.13 Depiction of the kinetics of segregation. At cold temperatures, the Ag lacks 

the mobility to segregate. At high temperatures, the Ag “floats” to the 

surface. At intermediate temperatures, a compositional gradient exists. . . . 110 

4.14 Depiction of a superlattice unit cell using the parameters in equation (4.34). 111 

4.15 High-angle x-ray diffraction peaks for a Ag/Ni multilayer with � = 4.2nm. 

The order of the satellite is given above its peak. The data taken at E = 

8320 eV are offset from the E = 8020 eV data. ...............112 

4.16 An illustration of a reciprocal space map along qx and qz. The specular 

ridge samples along qz, transverse scans sample along qx, and longitudinal 

scans sample both components. ........................114 

4.17 Specular reflectivity profiles (a) and longitudinal diffuse scattering profiles 

(b). The tilt of the sample from the specular condition is indicated by �θ, 

and the x-ray spectra taken at E = 8020 eV and E = 8320 eV are offset. 

The data and fitted/simulated profiles are given by the open circles and 

solid (or dashed) curves, respectively. The solid curves show the results of 

a simulation with a Ag/Ni (surface side / substrate side) chemical roughness 

of 0.24 nm and a Ni/Ag chemical roughness of 0.30 nm. The offset dashed 

curves show the results of a simulation with a Ag/Ni chemical roughness 

of 0.30 nm and a Ni/Ag chemical roughness of 0.40nm............115

List of Figures xii 

4.18 (a) A schematic showing the width between Ag and Ni (111) planes at 

the interface. (b) The interface width for Ag/Ni and Ni/Ag interfaces. 

The mean (111) d-spacing, 0.2197 nm, is indicated by the horizontal line. 

(d111,Ag = 0.23592 nm, and d111,Ni = 0.20345 nm). .............118 

4.19 Schematic showing the method by which SUPREX treats gradients in the 

out-of-plane d-spacing. ............................119 

4.20 High-angle x-ray diffraction peaks for the � = 3.2nm(a) and � = 5.9nm 

(b) Ag/Ni multilayers. The intensity axis is on a logarithmic scale, and the 

data taken at E = 8320 eV is offset above the E = 8020 eV data. The 

circles are the measured intensities, the solid curve is the fit for the strain 

gradient model, and the dotted curve is the best fit when strain gradients 

are disallowed. The order of the reflection is indicated above the peak. . . . 119 

4.21 Calculated {111} d-spacing profiles for the � = 3.2nm(a) and 5.9nm(b) 

Ag/Ni multilayers. The results of Jaszczak et al. for their theoretical multilayer 

are provided in (c) [113]. The vertical dashed lines are the strain-free 

d-spacings for the bulk phases. ........................120 

A.1 Diagram showing the relationship between the sample coordinate system, 

S, and the laboratory coordinate system, L. .................146 

B.1 The focusing circle. ..............................156 

D.1 The ratio of ɛ ′ 33 /ɛ′ D.2 

11 in the Vook-Witt model. .................172 

The biaxial modulus in units of 100 GPa using the Vook-Witt model. ....173 

D.3 The maximum resolved shear stress on the (111) slip plane given in GPa 

using ɛ ′ 11 = 0.704%. .............................174

List of Tables 

1.1 Ability of wafer curvature and x-ray techniques to measure different types 

of stresses. Adapted from Ruud [195]. . ................... 11 

1.2 Parameters for ion beam sputter deposition and substrate cleaning. ..... 18 

2.1 Measured and calculated values of the Young modulus, E, and the biaxial 

modulus, Y , for both free-standing and substrate-bonded films on the order 

of 1 µm thick. . . . .............................. 20 

2.2 Parameters for a-SiNx reactive sputter deposition. .............. 26 

2.3 Thermomechanical properties of the substrates used for the thermal cycling 

experiments. .................................. 27 

2.4 Calculated values of the biaxial modulus for selected films. The standard 

deviation is ±22GPa.............................. 35 

2.5 Measured values of the biaxial modulus for films of thickness 1 µm or greater. 38 

2.6 Comparison of the measured and calculated values of the biaxial modulus 

for the as-deposited films. ........................... 43 

2.7 Comparison of the measured and calculated values of the biaxial modulus 

for the annealed films. ............................. 44 

4.1 The Ag/Ni {111} Interface Energy . . . ................... 96 

4.2 {111} Interface Stress Results for FCC Metallic Multilayers . ........ 97 

xiii

Acknowledgments 

The acknowledgments is the only section anyone ever reads. That means I should have 

started writing it long ago, rather than waiting until the eleventh hour. I would have loved 

to make it witty and thoughtful, but there’s no time. Plus, for some reason, it’s not very 

easy to write. I hope nobody reads the acknowledgments simply to see what I thought of 

him or her. I love everybody here — it’s been great. 

First and foremost, thanks to Prof. Frans Spaepen, my advisor. Anyone who meets him 

knows how intelligent he is. Having been his student, I also know how great he is as a 

person. I’m honored to have been his advisee. 

Thanks to Profs. Mike Aziz and John Hutchinson. Along with Frans, these two demon- 

strate that it’s possible to be a brilliant scientist, a good teacher, and a nice person, proving 

that these traits are not necessarily mutually exclusive. 

Thanks to Dr. Denis McWhan. Without him, there would have been no Ag/Ni project. 

He taught me everything I know about synchrotrons and gave me a glimpse of the way sci- 

ence is done outside academia; plus, he gave me that near-death experience people some- 

times talk about. Well okay, it wasn’t quite near-death, but it had potential. Thanks for 

keeping your cool, Denis. 

Thanks to Prof. David Wu. He’s another great guy I’m honored to know. Despite what 

I’ve told him, I hope he never changes. Science needs more people like him. 

Thanks to Prof. David Turnbull. I don’t feel like I know him very well, but he’s my 

hero, nonetheless. He’s the epitome of a scientist, as well as an outstanding human be- 

ing. Everyone in the materials sciences, and especially those at Harvard obviously owe a 

tremendous amount to him. 

Thanks to the staff here at Harvard. There’s Yuan Lu, Warren MoberlyChan, John

Acknowledgments xv 

Chervinsky, and others. They’re all very good at what they do, and they’re good people, to 

boot. And, of course, there’s Frank Molea: to me, Frank represents everything I love about 

the fourth floor of McKay, and I’m proud to know him and call him my friend. I wish the 

best for him. 

Thanks to Denis Yu. Denis is the most selfless guy I’ve every known; doing things 

above and beyond the call of duty/friendship is normal for him. I’m embarrassed to think 

of the number of times he’s bailed me out. Good luck to you, Denis. Please let me know if 

there’s any way I can possibly pay you back. 

Thanks to Andrea Del Vecchio and Jen Sage; they’ve been great friends and office- 

mates. Andrea and Vince have been my surrogate family this past year. Jen also watches 

out for me by bringing back leftovers from the Women in Physics meetings. Thanks also 

to Anita Bowles, Jason Draut, Jeff Warrender, Bola George, and Cheng-Yen Wen. I can’t 

think of a more caring group of individuals. Special mention goes to Cheng-Yen. He took 

all the TEM images used in this thesis. He’s some sort of TEM prodigy, if such a thing 

exists. My wish for him is that he doesn’t get stuck doing TEM for everybody like he did 

for me. 

Thanks to the post-docs: John Leonard, Alex Cuenat, Manoj Pillai, Yuechao Zhao, and 

Joris Proost. With the exception of John, this club of post-docs is like my guide to the 

world outside the states. Interestingly, John is the guide to everywhere within the states. 

I’ve enjoyed chatting with all of you. 

The dynamics of the lab change every year. I could go through all the names of past 

students and post-docs, but it’s really pointless since I know they won’t come back to read 

my silly acknowledgments. However, in case they do, I thank them, too.

Acknowledgments xvi 

I would be remiss, however, if I didn’t mention Tim Van Rompaey. He worked on Ni/Ti 

multilayers for a summer over here, and he’s a great guy. I hope he’s enjoying graduate 

school as much as I did. 

My mom and dad have been terrific. I might as well thank them here for shaping me 

into who I am. Everyone I meet and talk to affects me, but these are only higher order 

effects. Thank you, Mom and Dad. 

Finally, thanks to Jessica for loving me unconditionally. I’m honestly not sure why she 

does, but I’m all the happier for it. I hope we’ll spend the rest of our lives together.

Dedicated to my family

Chapter 1 

Background 

This thesis deals with the measurement of stresses in thin films and multilayers using 

two different techniques: wafer curvature and x-ray diffraction. The techniques are com- 

plementary in that wafer curvature measures the average biaxial stress in the film, while 

x-ray diffraction measures the strain within the diffracting grains. This strain can be con- 

verted into a stress if the reference lattice parameter and the elastic constants of the grain 

are known. 

In Chapter 2, I will discuss the elastic properties of copper thin films. Our interest is not 

in the “true” modulus that depends on the curvature of the potential energy function and is 

usually measured by dynamic methods, but in the macroscopic and static modulus that is 

sensitive to material defects. We find the modulus of copper films is reduced relative to the 

dynamic value; it also falls off more quickly than the dynamic value at higher temperatures 

(≈ 0.4 × the melting temperature). Anelastic mechanisms appear to contribute to the 

modulus reduction. 

In the next section, I describe our experiments on the plastic behavior of copper thin 

1

Chapter 1: Background 2 

films on rigid substrates. While “smaller is stronger” is now an old saw in the thin film 

mechanics world, why it is so remains an issue of debate. A materials scientist looks at 

the problem from a dislocation standpoint; a mechanician considers constitutive equations. 

Strain gradient plasticity (SGP) merges these viewpoints and thus is a mathematically and 

physically satisfying theory. It is still a theory under development, however, and requires 

additional experimental support. Although we were unable to verify one of the predictions 

of SGP in thin films, our experiment can be used as a starting point for future attempts. 

Finally, we will deal with interface stresses in Ag/Ni multilayers. This field is also 

tinged with controversy, although not for a lack of good reason. J. W. Gibbs died before 

formulating the thermodynamics of solid-solid interfaces, and we’ve been struggling ever 

since. We show that the interface stress of Ag/Ni {111} interfaces is compressive, i.e., the 

interfaces push out. This is slightly counterintuitive, but the compressive nature results 

from higher order effects in the interatomic potential. (Our intuition is usually only good to 

first order). I also show that our method of measuring the interface stress is indeed valid for 

Ag/Ni; for other multilayers, complications such as intermixing can affect the measurement 

and should not be ignored. 

Since we will discuss stresses and strains in thin films, this chapter will review some 

of the essential mechanics and thin film basics (growth and microstructure). Subsequently, 

I will describe the wafer curvature and x-ray method of extracting stresses and strains, as 

well as explain how we grow our films and multilayers. 

1.1 Elasticity 

In what follows, I will only consider “stretch” deformations and no rotations.


The elastic strain, i.e., the time-independent reversible strain, results from atoms resid- 

ing at high energy positions away from the equilibrium spacing, a0. Figure 1.1 (a) shows 

two pair potentials; the solid line corresponds to a common nonconvex pair potential, the 

Lennard-Jones [140], and the dashed line corresponds to a harmonic potential. Their asso- 

ciated first and second derivatives are depicted in Figure 1.1 (b) and (c). Wefirst consider 

the nonconvex potential. 

By expansion around the equilibrium interatomic spacing, the energy per unit volume 

of a crystal can be expressed as 

w = w0 + 1 

2! cijklηijηkl + 1 

3! cijklmnηijηklηmn + ... (1.1) 

where the cs are nth order elastic constants corresponding to the nth order derivatives of 

energy and the ηs are Lagrangian strains. (Appendix F has a list of symbols used in the 

thesis). They are defined in terms of the displacement vector u by 

ηij = 1 � � 

ui, j + u j,i + uk,iuk, j . (1.2) 

2 

The Lagrangian strains are referenced to the undeformed state, as opposed to Eulerian 

strains, which are referenced to the deformed state. The thermodynamic tensions, Tij, are 

then defined by the differential of strain energy with respect to the Lagrangian strains: 

dw = Tij dηij. (1.3) 

If all the components of the displacement gradient are small, i.e., |ui, j| ≪1, the La- 

grangian strains reduce to 

ɛij = 1 � � 

ui, j + u j,i , (1.4) 

2


σ = F/a 0 2 

Figure 1.1: Interatomic potentials (a) and their first (b) and second (c) derivatives. Both 

the Lennard-Jones potential (solid) [140] and a harmonic potential (dashed) are shown.


and the Hookean stresses are defined by 

dw = σij dɛij. (1.5) 

For the small strain assumption, the distinction between the undeformed and deformed state 

is unnecessary. 

Making the small strain approximation is tantamount to choosing a harmonic potential 

(the dashed curve in Figure 1.1 (a)). The stress then varies linearly with strain, as shown 

in Figure 1.1 (b), and we may write 

σij = cijklɛkl (1.6a) 

or ɛij = sijklσkl (1.6b) 

depending on which is the independent variable. For the one-dimensional model depicted 

in Figure 1.1, c1111 is independent of strain (Figure 1.1 (c)). 

In addition to writing the stress in terms of strain, I can also write stress in terms of its 

definition of force per unit area: 

σij = Fi 

A j 

where the area vector points along the plane normal. 

(1.7) 

A material’s measured elastic constants may differ substantially from those calculated 

from the curvature of the potential well. This may result from either point, line, areal, or 

volume defects in the material. If the properties are time-dependent and reversible, they are 

anelastic. The reversible motion of defects can cause an abnormally large strain, thereby 

reducing the measured elastic constant. This phenomenon may or may not be measurable 

depending on the time scale of the experiment since the defect kinetics have characteristic 

time constants.


For more information on linear elasticity, the texts by Landau and Lifshitz [136], Timo- 

shenko and Goodier [224], and Sokolnikoff [210] are recommended. Finite-strain elasticity 

is discussed by Murgnahan [169] and Gurtin [91], and details on the higher order elastic 

constants can be found in the seminal papers by Wallace [240] and Brugger [23]. The 

classic text on anelasticity is by Nowick and Berry [175]. 

1.2 Dislocation Plasticity 

Plastic flow is time-dependent and irreversible. Here, I focus on the subject of dis- 

location plasticity and on one particularly important equation relevant to the strength of 

materials. 

Arzt [6] emphasizes the strong coupling between characteristic lengths and size pa- 

rameters in determining the material strength. The characteristic length is defined as the 

dimension associated with the relevant physical process (such as the Burgers vector or the 

equilibrium diameter of a dislocation loop). The size parameter is related to the microstruc- 

ture (such as the grain size or dislocation spacing in a forest dislocation network). A simple 

example following Orowan is given to show how a relation between the Burgers vector and 

obstacle spacing can determine material strength. 

tion: 

An applied shear stress, σ , imparts a Peach-Koehler force per unit length on a disloca- 

F = σ b (1.8) 

where b is the Burgers vector. The dislocation line tension, T , resists bowing as shown in


Figure 1.2 (a). The line tension is given by 

T ≈ 1 

2 Gb2 

where G is the shear modulus. Balancing the vertical forces shown in (a) gives 

(1.9) 

σ b (2�R) = 2T � (1.10) 

where the length of the line segment is 2�R, and we assume � is small. Thus, if a moving 

dislocation encounters the two obstacles shown in Figure 1.2 (b), the shear stress needed to 

bow the dislocation to the maximum curvature of κ is determined by the obstacle spacing, 

ℓ = 2/κ: 

σOr ≈ Gb 

. (1.11) 

ℓ 

Bowing the dislocation beyond this critical radius requires no additional stress; thus, σOr is 

the maximum resistance to the applied shear stress for the case shown in Figure 1.2 (b). 

Assuming that these obstacles, whether they are other dislocations or precipitates, rep- 

resent the overall maximum resistance to dislocation motion in a material, the Orowan 

stress, σOr, represents the strength of a material. This ignores the problem of thermal acti- 

vation as described by Kocks et al. [130]. 

Strain gradient plasticity merges dislocation mechanics with continuum mechanics. 

The importance of plastic strain gradients had been recognized by Nye, who formulated 

the idea of geometrically necessary dislocations in association with lattice curvature [178], 

prompting mechanicians to develop a continuum description of lattice curvature with a non- 

Riemannian space formulation [15, 132, 133]. Every dislocation present in a material can 

be categorized as either a geometrically necessary dislocation (GND) required for defor- 

mation compatibility or a statistically stored dislocation (SSD) [43] that is trapped by other


T 

T sinΘ 

σb (2ΘR) 

R 

Θ Θ 

T 

T sinΘ 

(a) (b) 

Figure 1.2: (a) A depiction of the vertical force balance between dislocation line tension 

and the Peach-Koehler force. (b) An illustration of a dislocation bowing between two 

impenetrable obstacles with separation ℓ. 

dislocations in a random way. Ashby [7, 8] formulated a more general theory of GNDs and 

emphasized their importance in determining a material’s plastic properties. 

According to Ashby, the density of the GNDs, ρGND, depends on the geometrical ar- 

rangement of grains and phases and is therefore microstructure-dependent. On the other 

hand, the density of SSDs, ρSSD, depends on the material properties and the total plastic 

strain. The total dislocation density is approximately (ρGND + ρSSD), although ρGND can 

increase ρSSD rapidly at large strains [8]. 

and 

The densities of GNDs due to edge and screw dislocations are given by [5] 

ρ edge 

GND = −∇ɛsh · s 

b 

ρ screw 

GND = ∇ɛsh · m 

b 

l 

(1.12) 

(1.13)


where ɛsh is the plastic shear strain, s is a unit vector along the slip direction, and m is given 

by m = s × n (where n is the slip plane unit normal). Ashby [7, 8] makes use of a simpler 

relation, 

ρGND = 4ɛsh 

ℓGNDb 

where ℓGND is called the geometrical slip distance characteristic of the microstructure. 

(1.14) 

If we assume a Taylor model of strain hardening [216, 217] (see Chapter 3 for details), 

the flow stress is given by 

σflow = kTGb √ ρSSD + ρGND 

(1.15) 

where kT is a constant on the order of 0.1 to 1. By combining (1.14) and (1.15) and 

assuming ρGND ≫ ρSSD, we see that for cases in which the local shear strain is proportional 

to b/ℓGND, the Orowan stress equation is recovered. 

By inserting a gradient of plastic shear into the strength problem, Ashby opened up the 

door to mechanicians who had the tools to develop continuum theories of plasticity that 

can incorporate a characteristic length scale. Probably the most well developed of these is 

by Fleck and Hutchinson [61]. They extended the J2 flow theory to include both stretch 

gradients and rotation gradients, each of which can be represented by a length parameter. 

For more information, Cottrell’s books, The Mechanical Properties of Matter [43] and 

Dislocations and Plastic Flow in Crystals [42] serve as excellent introductions to plasticity 

from a materials science viewpoint. Argon’s recent reviews in [3, 4] summarize the current 

view of the field. Hill’s Mathematical Theory of Plasticity is the usual starting point for a 

mechanician [99].


1.3 Thin Film Growth and Microstructure 

When a depositing atom arrives at a substrate of a different phase, its incorporation 

depends on its incoming energy, the overall deposition rate, the temperature of the substrate, 

the defect structure of the surface, etc. [25, 230]. If the mismatch with the layer below it 

is small, nucleation and growth may proceed pseudomorphically, and the interface may 

be described as coherent if the lattice registry is complete [104]. The strain energy of 

the film grows linearly with the film thickness; consequently, it may become energetically 

favorable for a periodic array of misfit dislocations to form at the interface which maintains 

the crystallographic registry between them. Such an interface is termed semicoherent. If 

the mismatch is very large, the registry of the atomic planes is lost, and the interface is 

incoherent. 

Non-epitaxial films can develop texture and undergo grain growth during deposition 

[127, 220]. The substrate temperature is often the most important parameter in determining 

the film microstructure. This has led to the development of structural zone models that 

predict grain sizes and shapes as a function of substrate temperature [156, 145, 144]. 

1.4 Two Common Stress Determination Techniques 

Equations (1.6) and (1.7) show that stresses may be determined by both strain and force 

measurements. Ideally, both measurements are taken simultaneously during a deformation; 

this allows an evaluation of the elastic constants [107]. Unfortunately such techniques 

are not often performed on thin film samples. The most common force measurement for 

thin films on substrates is the wafer curvature measurement. The equation used to extract


Table 1.1: Ability of wafer curvature and x-ray techniques to measure different types of 

stresses. Adapted from Ruud [195]. 

Stress Type Wafer Curvature X-Ray 

Growth Yes Yes 

Thermal Yes Yes 

Coherency No Yes 

Interface Phase Formation Yes No 

Interface Stress Yes No 

the stress from the substrate curvature is derived from force and moment balance. X-ray 

diffraction is the most common way of making strain measurements. 

Both of these techniques benefit from their ease in experimental setup and analysis. 

They may not give the same results, however. Following Table 1.1, the wafer curvature 

method can be used to measure the following stresses: growth stress, thermal stress, inter- 

face stress, and stresses associated with phase formation at the interface. X-ray diffraction 

can be used to measure growth stress, thermal stress, and coherency stress. Thus, we can 

isolate certain stresses by using both techniques on the same sample. 

1.4.1 Wafer Curvature 

Figure 1.3 (a) shows a strain-free free-standing film and substrate. Forcing the film to 

fit onto the substrate requires an applied force as shown in (b), and the film is subsequently 

“glued” onto the substrate (c). Force and moment balance require that the substrate bends 

to accommodate the film stress (d). 

The relationship between the substrate curvature change, �κ, and the film stress, σf, is 

given by Stoney’s equation: 

σf = Yst 2 s �κ 

6tf 

(1.16)


(a) (b) 

(c) (d) 

Figure 1.3: A schematic showing how the film stress causes substrate curvature. (a) The 

film and substrate are both shown in their strain-free state. (b) The film is stretched to fit 

onto the substrate. (c) The film is “glued” onto the substrate. (d) The film/substrate system 

bends in order to achieve force and moment balance.


Figure 1.4: A schematic of the curvature measurement apparatus at Harvard University. 

Adapted from Mullin [158]. 

where Ys is the biaxial modulus of the substrate, ts is the substrate thickness, and tf is the 

film thickness. This equation assumes tf ≪ ts; in this limit, determination of the film stress 

does not rely on knowledge of the biaxial modulus of the film. Appendix E provides a 

derivation of Stoney’s equation. 

The wafer curvature measurement / annealing apparatus at Harvard University is called 

ROC and is shown in Figure 1.4. The technical details of how it measures the curvature can 

be found in Witvrouw’s or Mullin’s thesis [256, 158], but the principle is well known [62]. 

The 1 inch by 1/4 inch sample rests in a furnace on two sapphire prisms and is abutted on 

two adjacent sides by steel pins to allow repeatable sample placement. A HeNe laser scans 

across the long axis of the sample in 1 mm steps, and the reflected beam is collected in a 

null detector. The position of the reflected spot is recorded as a function of position along 

the sample, thereby giving the sample curvature.


Prior to thermal cycling, the chamber is evacuated to a pressure of 3 × 10 −4 Pa by a 

diffusion pump and then backfilled to a slight overpressure of 3 psi with a flowing gas of 

95 wt.% He and 5 wt.% H2. 

1.4.2 X-Ray Diffraction Measurements 

As mentioned, strain measurements are often performed by x-ray diffraction, with the 

strain-free lattice parameter, a0, and the elastic constants known a priori. Although this 

measurement is straightforward in principle, it still has several problems. First, not every- 

one agrees on the value of a0. Its value is often determined by performing x-ray diffraction 

experiments on a strain-free sample, e.g., an annealed powder. Extremely precise tech- 

niques have been developed to minimize the errors of these measurements [60]. In practice, 

however, a completely strain-free sample is difficult to produce. Furthermore, the meaning 

of “strain-free” can be different depending on what the researcher is trying to measure. It 

may refer to the constrained zero point between an applied tension and compression, or, it 

may refer to equilibrium lattice parameter. These may differ depending on the thermody- 

namic state of the material. 

The determination of elastic constants in polycrystalline materials is also problematic. 

There exists a substantial body of literature devoted to the proper ways of averaging the 

single crystal elastic constants for textured and untextured samples [24]. In principle, one 

can determine a sample’s orientation distribution function (ODF) and use a grain interaction 

model (e.g., Voigt or Reuss) to determine the stresses from the measured strains. ODF 

determination is a tedious process, however, and researchers typically either assume the 

sample has fiber texture or no texture at all. The grain interaction model is also often


assumed. 

Appendix A provides some details on how x-ray strain measurements work. For more 

information, the classic text for stress determination by x-rays is the book by Noyan and 

Cohen [176]; there are several errors in this book that are corrected in Appendix B. Noyan 

et al. [177] have also written a review on x-ray measurements of thin film stresses, and 

Hauk et al. [95] have recently completed a comprehensive review of techniques used to 

measure residual stresses. 

A schematic of the diffractometer used at Harvard is shown in Figure 1.5. It consists 

of a four-circle Huber diffractometer (a Huber 424 two-circle with a 511.1 Eulerian cra- 

dle) equipped with a Model 20/20 linear position sensitive detector (PSD) from Reflection 

Imaging (www.reflectionimaging.com). The system was put together with the 

help of Molecular Metrology (www.molmet.com). Since the pinhole collimator has a 

diameter of 0.3 mm, the samples are rocked ±2 ◦ in the diffractometer plane to achieve 

adequate sampling statistics [111] (Appendix B). 

Beamline X22C at the National Synchrotron Light Source at Brookhaven National Lab- 

oratory was used for reflectivity [194], θ − 2θ, and grazing incidence diffraction (GID) 

[148] measurements. The diffractometer is a six-circle Frank and Heydrich, and the inci- 

dent beam is conditioned with a Pt focusing mirror and a two-crystal Si(111) monochro- 

mator. 

A schematic of the different scattering geometries is presented in Figure 1.6. Figure 

1.6 (a) shows the symmetric θ − 2θ geometry used for x-ray reflectivity or powder scans. 

As indicated by the arrow on the left, χ-scans are performed by rotating the sample normal 

into or out of the plane of the page while maintaining the diffraction condition. Figure


Figure 1.5: A schematic of the four-circle Huber diffractometer at Harvard University. 

1.6 (b) shows the longitudinal scan geometry used in diffuse scattering (Chapter 4). The 

sample is tilted slightly away from the specular condition by an amount �θ. The geometry 

for grazing incidence diffraction is depicted in Figure 1.6 (c); here, the scattering vector is 

tilted with respect to the sample surface by a small angle α. The magnitude of the scattering 

vector, q,isgivenby 

q = 

4π sin θ 

λ 

where λ is the x-ray wavelength and 2θ is the diffraction angle. 

1.5 Sputter Deposition 

(1.17) 

The majority of the films used in this thesis were grown using the sputtering chamber 

built by Spaepen et al. [213] and shown in Figure 1.7. Prior to deposition, 1 inch by 1/4 inch


χ 

θ−∆θ 

2θ 

θ 

α 

q 

∆θ 

n 

Figure 1.6: A schematic of different scattering geometries. (a) The symmetric θ − 2θ 

geometry and χ rotation axis, (b) the geometry used for longitudinal diffuse scattering, 

and (c) the grazing incidence diffraction geometry. 

θ 

θ+∆θ 

α 

(a) 

(b) 

(c)


Figure 1.7: A schematic of the sputtering chamber at Harvard University. 

substrates were cleaved from the wafer and subsequently cleaned in acetone, methanol, and 

isopropanol. 

The base pressure of the deposition chamber is 1.3 × 10 −5 Pa. The sputtering targets 

are bolted onto a rotating target block to facilitate multilayer growth, and both the targets 

and samples are water-cooled. One ion gun is used for target sputtering, and another gun 

can be used for either substrate cleaning or reactive sputtering. 

The parameters for sputter deposition and substrate cleaning are listed in Table 1.2. 

Table 1.2: Parameters for ion beam sputter deposition and substrate cleaning. 

Deposition Substrate Cleaning 

Beam 1400 V 45 mA 500 V 35 mA 

Accelerator 150 V 0 mA 500 V 0 mA 

Discharge 25 V 2 mA 30 V 2 mA 

Cathode 8 V - 8 V - 

Ar Pressure 1.1 × 10 −2 Pa 1.1 × 10 −2 Pa

Chapter 2 

The Elastic Properties of Cu Thin Films 

Using a tensile tester especially designed for free-standing films, Huang [106] found 

that the Young moduli of electron beam evaporated films were about 20% lower than those 

calculated from the elastic constants found in the literature (such as those from Landolt- 

Börnstein [12], Simmons and Wang [207], or Huntington [108]). Table 2.1 shows that this 

“deficit” is well documented and only observed in polycrystalline films. 

A variety of reasons for this modulus deficit have been hypothesized. They include 

elastic anisotropy (from preferred orientation), porosity, grain boundary compliance, grain 

boundary cracking, reversible microplasticity, dislocation anelasticity, and grain bound- 

ary sliding [107]. We studied the origin of the modulus deficit by measuring the biaxial 

modulus of sputtered copper films on substrates using the wafer curvature method. Before 

discussing how the measurement is made, it is important to first understand the stress- 

temperature profile. 

19

Chapter 2: The Elastic Properties of Cu Thin Films 20 

Table 2.1: Measured and calculated values of the Young modulus, E, and the biaxial modulus, 

Y , for both free-standing and substrate-bonded films on the order of 1 µm thick. 

Material Emeas E 〈111〉 

calc ∗ Ymeas Y 〈111〉 

calc ∗ (GPa) (GPa) (GPa) (GPa) 

Preparation 

Method 

Ref. 

Ag/Cu 87.5 107 - - e-beam [106] 

Cu 102 130 - - e-beam [106] 

Cu 104 130 - - e-beam [187] 

Cu 110 130 - - sputtered [187] 

Cu 66 130 - - electrodep. [187] 

Cu 104 130 161 261 sputtered [262] 

Cu 116 130 - - electrodep. [192] 

Ag 63 84 - - e-beam [106] 

Ag - - 50 174 thermal evap. [46] 

Al 57 72 - - e-beam [106] 

Al 49 72 - - thermal evap. [97] 

Al 64 72 - - sputtered [192] 

Al - - 62 115 thermal evap. [46] 

W 340 389 - - CVD [192] 

Si (110) † 166 166 - - etched from bulk [215] 

∗ Calculated from the dynamic elastic constants found in Bechmann et al. [12] 

† Single crystal


Figure 2.1: Stress as a function of temperature for a 1.91 µm sputtered Cu film on Si(100). 

The dotted lines show the hypothetical elastic stress in the film if no relaxation could take 

place (thermoelastic lines), and the labels (A), (B), and (C) denote elastic thermal cycles 

as described in the text. The upper thermoelastic line was drawn using the elastic constant 

data found in Bechmann et al. [12] and assuming mild 〈111〉 texture. 

2.1 The Variation of Stress with Temperature in a Thin 

Film on an Elastic Substrate 

As shown in Figure 2.1, the initial stress in the film is compressive due to atomic peen- 

ing [254]. (Tensile stresses are always plotted as positive values in this thesis). Due to the 

thermal expansion mismatch with the substrate, the copper film is compressed upon heat- 

ing and stretched upon cooling (cf. Figure 1.3). For small temperature changes �T , the 

deformation is elastic, and cycling back and forth within a small temperature range allows 

a measure of the biaxial modulus, Yf, if the thermal expansion coefficients of the substrate


and film (αs and αf) are known since 

�σf = Yf�ɛf (2.1) 

= Yf 

� T1 

T0 

dT (αs − αf) 

≈ Yf(αs − αf)�T. 

These small temperature cycles are called elastic thermal cycles from now on. Figure 2.1 

shows two elastic thermal cycles from 10 ◦ Cto70 ◦ C for the as-deposited and the annealed 

film (labeled (A) and (B), respectively). 

Compressing or stretching beyond the elastic limit induces plastic deformation by a 

mechanism that depends on the temperature. If the thermal cycle is started higher than 

some critical temperature (typically about 300 ◦ C for most of the sputtered films), nearly all 

the strain is plastic. An example of this is shown in Figure 2.1, where an “elastic” thermal 

cycle (C) from 500 ◦ C to 550 ◦ C demonstrates hysteresis and is therefore not elastic. Below 

the critical temperature, it is possible to obtain elastic deformation and hence to determine 

the biaxial modulus as a function of temperature. This so-called critical temperature is 

dependent on the heating/cooling rate, stress state, and the mechanical properties of the 

film. 

Several researchers have attempted to explain the shape of the σf vs. T profile [54, 124, 

125, 172]. The initial ramp-up in temperature from the as-deposited state causes a large 

tensile increase at approximately 200 ◦ C. Chaudhari [35] has shown that this is due to grain 

growth. (A refinement of his model has recently been published [58]). A further increase 

in temperature results in relaxation of the stress by dislocation flow and atomic motion. 

Holding at any of these temperatures produces stress relaxation that can be described by 

various creep models [3, 4, 6, 69].


Again following Figure 2.1, atomic motion fully relaxes the film at 550 ◦ C,andanew 

strain-free lattice parameter is established. Cooling from 550 ◦ C puts the film in tension. 

As the film cools, it becomes increasingly difficult to relax the stress: thus, the residual 

stress builds up. The resulting stress at room temperature is dependent on the temperature 

at which the strain-free lattice parameter is established and how difficult it is for the film to 

undergo plastic relaxation. The dotted lines in Figure 2.1 illustrate the stress-temperature 

behavior the film would take if no plastic flow were possible. At any given temperature, the 

difference between the dotted line and the measured residual stress indicates the amount of 

plastic strain in the film. After the first thermal cycle takes place, the shape of the stress 

vs. temperature profile remains constant. Indeed, it is now commonly accepted that the 

microstructure is stable after heating through the grain growth phase of the first thermal 

cycle [10]. 

It should be noted that this stress-temperature profile changes dramatically when the 

films are prepared and annealed in ultra-high vacuum conditions rather than in high vacuum 

[246]. 

2.2 The Biaxial Modulus Measurement 

As equation (2.1) shows, the biaxial modulus of the film can be determined if the coef- 

ficients of thermal expansion (CTE) of the film and substrate are known. 

Alternatively, if elastic thermal cycles are performed on different substrates with known 

thermomechanical constants, it is possible to determine both the biaxial modulus and the 

CTE of the film. For instance, if we deposit the same film on two different substrates 1 and


2, 


Yf = 

1 

(α1 − α2) 

� �σ1 

�T 

� 

�σ2 

− 

�T 

(2.2) 

αf = α2�σ1 − α1�σ2 

. (2.3) 

�σ1 − �σ2 

If more than two substrates are used, we can perform a linear fit of�σi/�T vs. αi where 

�σi is the change in film stress over the elastic thermal cycle for a substrate i. The slope 

of this line is equal to −Yf, and the intercept is αfYf. 

Note that pairing a substrate with a single CTE with a substrate with anisotropic CTEs, 

such as Y -cut quartz, adds yet another equation and can lead to an independent determi- 

nation of Young’s modulus, Poisson’s ratio, and the CTE of the film (assuming the biaxial 

modulus of the film is isotropic in the plane) [262]. 

2.2.1 The Effect of Texture on the Biaxial Modulus 

Copper exhibits strong elastic anisotropy; this can manifest itself if the polycrystalline 

film has preferred orientation, i.e., texture. The calculated value of the biaxial modulus 

is 261 GPa for 〈111〉 textured copper, 196 GPa for a randomly oriented polycrystal, and 

115 GPa for 〈100〉 texture (cf. Figure D.2 in Appendix D). Without effects from defects 

in the microstructure, the biaxial modulus of polycrystalline films is approximately the 

volume average over all the grain orientations: 

〈X〉 = 1 

8π 2 

� ϕ1=2π � �=π � ϕ2=2π 

X (ϕ1,�,ϕ2) f (ϕ1,�,ϕ2) sin � dϕ1 d� dϕ2, (2.4) 

ϕ1=0 �=0 ϕ2=0 

where X is either the biaxial modulus or the inverse biaxial modulus, f (ϕ1,�,ϕ2) is the 

orientation distribution function (ODF) and ϕ1, �, and ϕ2 are the Euler angles [24, 131].


See Appendix D for more on averaging techniques. This calculation does not take into 

account stress continuity at the grain boundaries [10]. 

2.3 Experiment 

2.3.1 Film Preparation 

A number of substrates were used: Si(100) with a 1 µm thermal oxide, bare Si(100), 

Ge(111), α-Al2O3(0001), and fused silica; Si(100) with the thermal oxide was used for 

most experiments. Before deposition, the baseline curvatures of the substrates were mea- 

sured. 

Copper films (from a 99.99 at.% Cu target) of thicknesses ranging from 0.2 µm to 

2.3 µm were deposited by ion beam sputtering with an Ar pressure of 1.1 × 10 −2 Pa after 

reaching a base pressure of 1.3 × 10 −5 Pa. The sputtering chamber described by Spaepen 

et al. [213] (Figure 1.7) was used with the sputtering parameters given in Table 1.2. 

For some substrates, a thin amorphous silicon nitride (a-SiNx) layer (nominally 40 nm 

thick) was put down by dual ion gun sputtering before depositing the copper to act as a 

diffusion barrier between the film and substrate. A nitride layer of the same thickness was 

also deposited onto the back of the substrate so as not to affect the curvature measure- 

ment. These deposition parameters are in Table 2.2. Submicron copper films were grown 

on thin (97 µm thick) Si(100) substrates. Prior to deposition, the substrates were sputter 

cleaned for 2-4 minutes. The substrates were also water cooled from the backside, leading 

to temperature rise of 10 ◦ C after three hours of deposition. A typical deposition rate was 

0.15 nm/s.


Table 2.2: Parameters for a-SiNx reactive sputter deposition. 

Argon Nitrogen 

Beam 1200 V 35 mA 600 V 11 mA 

Accelerator 200 V 0 mA 200 V 0 mA 

Discharge 30 V 1.2 mA 40 V 2 mA 

Cathode 8 V - 8 V - 

Pressure 8 × 10 −3 Pa 5.3 × 10 −3 Pa 

Additionally, some films were electron beam (e-beam) evaporated from 99.99 at.% Cu 

shot (see [106]) at rates of 1 nm/s. In this chamber, the substrate temperature rises to about 

90 ◦ C after two hours of deposition. These films were grown primarily for comparative 

purposes. In what follows, nearly all of copper films discussed were sputter deposited. 

2.3.2 Film Characterization 

After deposition, the thickness of the films was measured using a Tencor Alpha-Step 

200 Profilometer. For very thin films (less than one micrometer thick), the thickness was 

also measured with Rutherford Backscattering Spectrometry (RBS). 

Microstructural characterization was performed by both plan-view and cross-sectional 

transmission electron microscopy (TEM). Film purity was checked by RBS and energy 

dispersive spectroscopy (EDS) in the TEM. 

2.3.3 X-Ray Diffraction 

Symmetric θ − 2θ scans (also known as powder scans) and (111) fiber texture plots 

(the χ-scans described in Chapter 1) were performed in order to estimate the film texture 

[44, 126]. The powder scans sample only a small portion of reciprocal space but can show 

the relative strengths of the 〈111〉 and 〈100〉 textures. The (111) fiber texture plots are cuts


Table 2.3: Thermomechanical properties of the substrates used for the thermal cycling 

experiments. 

Material Ys (GPa) ∗ CTE (10 −6 K −1 ) † Thickness (µm) Ref. 

Si(001) 180.5 2.708 395 & 97 [249] 

Ge(111) 183.4 5.905 461 [188] 

α-Al2O3(0001) 608.4 5.313 257 [86], [249] 

fused silica 86.5 0.52 162 [203] 

∗ at 25 ◦ C 

† at 40 ◦ C 

through the (111) pole figure and are sufficient to describe fiber textures, although a full 

ODF eliminates the peak overlap sometimes present in fiber texture plots. 

The fiber texture plots were performed on the diffractometer shown in Figure 1.5. As 

discussed by Wcislak et al. [241], using a linear position sensitive detector to find the 

integrated intensity of the peak as a function of χ eliminates most of the corrections needed 

with a point detector [126, 135], but the thin film absorption correction (Appendix B) is still 

required. The volume fraction of 〈hkl〉-oriented grains is given by Baker et al. [10]: 

where I 111 

hkl 

fhkl = 

� χmax 

0 

� χmax 

0 

I 111 

hkl sin χ dχ 

I 111 

total 

sin χ dχ 

(2.5) 

is the scattered intensity from the 〈hkl〉-oriented grains in the (111) texture plot. 

2.3.4 Curvature Measurements 

The wafer curvature apparatus used in these experiments was described in Chapter 

1, and the curvatures were converted to stresses using the Stoney equation (1.16). The 

thermomechanical properties of the various substrates used are listed in Table 2.3. 

The films were typically heated and cooled at approximately 5 ◦ C/min in a forming gas 

environment. For the elastic thermal cycles, the heating and cooling rate was varied from


1 ◦ C/min to 6 ◦ C/min. The temperature dependence of the modulus was studied during the 

heating part of the thermal cycle (Figure 2.13). 

2.4 Results 

2.4.1 Microstructure 

Plan-view TEM micrographs are presented in Figure 2.2 (a) and (b) for the as-deposited 

and annealed 0.87 µm thick copper, respectively. The grain size increases from ∼ 40 nm 

to ∼ 500 nm, the latter being the average of a bimodal grain size distribution. The cross- 

sectional micrographs of these films are presented in section 3.4.1. They show that even 

after annealing at 600 ◦ C for 0.5 hrs, the grains did not yet reach the stable microstructure 

consisting of columnar grains with sizes on the order of the film thickness [219]. 

A cross-sectional micrograph of Cu grown on Ge(111) is shown in Figure 2.3. The 

grains are columnar and extend from the substrate to the surface. The in-plane grain size is 

∼ 50 nm. 

Both RBS and EDS show no impurities in the bulk of the copper. The argon level in 

the annealed films was below the detection limit of EDS system. 

2.4.2 Film Texture 

Figure 2.4 (a) compares the powder scans for copper grown on Ge(111), α-Al2O3(0001), 

and Si(100). All three demonstrate 〈111〉 texture, although the copper grown on Ge(111) 

clearly has the strongest texture. This can be better seen in the χ-scan in Figure 2.4 (b) 

which shows the sharpness of the 〈111〉 texture pole.


(a) 

(b) 

Figure 2.2: Plan-view TEM micrographs of 0.87 µm thick copper in the as-deposited state 

(a) and after annealing at 600 ◦ C for 0.5 hrs (b).


Figure 2.3: A cross-sectional TEM micrograph of 1.90 µm thick copper grown on Ge(111). 

Annealing the films at 600 ◦ C induces grain growth, as seen in the micrographs, and 

enhanced 〈111〉 texture, as seen in Figures 2.5 (a) (powder scan) and 2.5 (b) (χ-scan). The 

films annealed on α-Al2O3(0001) had much stronger 〈111〉 texture than those on Si(100) 

(Figure 2.6). The films grown on Ge(111) reacted with the Ge above 200 ◦ C, so the texture 

after annealing was not remeasured. 

Figure 2.7 shows that the 0.5 µm thick film had even stronger 〈111〉 texture in both 

the as-deposited and annealed states than the 1.25 µm thick film (cf. Figure 2.5 (b)). The 

as-deposited electron beam evaporated films had both 〈111〉 and 〈100〉 texture components 

(Figure 2.8). 

If we had the full ODF from several pole figures, we could calculate the biaxial mod- 

ulus of imperfectly textured films using equation (2.4). However, even with only χ-scans, 

we can still estimate the biaxial modulus as a function of the volume fraction of textured


χ 

µ 

µ α 

µ µ 

χ 

〈100〉 〈111〉 

(a) 

(b) 

Figure 2.4: (a) Powder diffraction x-ray spectra for sputter-deposited copper films on 

Ge(111) (dashed), α-Al2O3(0001) (dotted), and Si(100) (solid). (b) (111) fiber texture 

plot of copper films on Ge(111) (dashed) and Si(100) (solid).


χ 

χ 

µ µ µ 

〈100〉 〈111〉 

(a) 

(b) 

Figure 2.5: (a) Powder diffraction spectra, and (b) a (111) fiber texture plot for asdeposited 

and annealed copper films on Si(100).


µ µ µ 

µ µ 

Figure 2.6: Powder diffraction spectra for annealed copper films on Si(100) (solid) and 

α-Al2O3(0001) (dotted). 

χ 

χ 

〈100〉 〈111〉 

Figure 2.7: (111) fiber texture plot of as-deposited and annealed 0.50 µm thick copper 

films.


χ 

χ 

〈100〉 〈111〉 

Figure 2.8: (111) fiber texture plot of an electron beam evaporated copper film. 

grains and the width of the texture pole. Figures 2.9 (a) and (b) show how the biaxial 

modulus depends on the pole width for both 〈111〉 and 〈100〉 texture, assuming a Gaussian 

distribution of the orientations about the pole. We can then estimate the biaxial modulus as 

the volume-weighted average: 

Y ≈ frandomYrandom + f100Y100(σ100) + f111Y111(σ111) (2.6) 

where f is the volume fraction of properly oriented grains, and σhkl is the standard de- 

viation of the texture pole. The results are presented in Table 2.4. All the calculations 

presented in this chapter are based on the Hill average, i.e., the arithmetic average of the 

Voigt and Reuss averages [100].


Figure 2.9: The calculated biaxial modulus for 〈111〉 texture (a) and 〈100〉 texture (b) as a 

function of the texture pole standard deviation. A Gaussian texture pole is assumed. 

Table 2.4: Calculated values of the biaxial modulus for selected films. The standard deviation 

is ±22 GPa. 

Substrate Preparation Method Thickness (µm) Y (GPa) Y (GPa) 

(as-dep.) (ann.) 

Si(100) sputtered 1.25 226 256 

Si(100) sputtered 0.50 237 256 

Si(100) e-beam evaporated 0.94 203 - 

Ge(111) sputtered 1.87 253 -


2.4.3 Biaxial Modulus 

Measuring Both the Biaxial Modulus and the CTE 

As mentioned in section 2.2, by using two or more substrates with different thermo- 

mechanical constants, one can extract both the biaxial modulus and the CTE of the film. 

We did this using the as-deposited films grown on Si(100), α-Al2O3(0001), and fused sil- 

ica, choosing these because the textures appeared to be similar. Since the sharpness of 

the 〈111〉 texture increases much more for copper on α-Al2O3(0001) than for Si(100) after 

annealing, the modulus after annealing could not be measured with this method. Using 

equations (2.2) and (2.3) for the data shown in Figure 2.10, we obtain Y = 156 ± 18 GPa 

and α = 20.2 ± 0.5 × 10 −6 K −1 .Ifwefit a line using only two substrates, Si(100) and α- 

Al2O3(0001), we arrive at Y = 179±32 GPa and α = 18.2±0.8×10 −6 K −1 . If we only use 

the Si(100) substrate and assume the bulk value of the CTE for Cu, α = 16.8 × 10 −6 K −1 , 

we find Y = 197 ± 8 GPa. The latter value is judged to be the correct one since the 

bulk CTE for a pure material should be insensitive to microstructural defects. The ther- 

momechanical constants of fused silica and α-Al2O3(0001) are not as well established as 

that of Si (particularly the former; see Mullin’s thesis for problems associated with fused 

silica [158]). Most importantly, the textures of the copper films on fused silica and α- 

Al2O3(0001) are slightly different from that on Si(100). Using the bulk CTE for copper 

and the thermomechanical properties in Table 2.3, we obtain Y = 184 ± 8 GPa for copper 

on fused silica, Y = 197 ± 8 GPa for copper on Si(100), and Y = 201 ± 8 GPa for copper 

on α-Al2O3(0001), the variation of which can easily be accounted for by slight texture or 

microstructural differences caused by the substrate. Thus, all the biaxial moduli described 

hereafter were obtained using the bulk CTE of copper.


∆σ ∆Τ 

α 

α × 10 −6 [Κ −1 ] 

Figure 2.10: Plot showing how the biaxial modulus and CTE can be determined for a set 

of different substrates. 

As-deposited and Annealed Films at Room Temperature 

Table 2.5 lists the average biaxial moduli for all the copper films with thicknesses 

greater than 1 µm. The standard deviation of ±8 GPa stems from the uncertainty in film 

thickness as measured by the profilometer. The annealed films refer to samples heated to 

550 ◦ Cor600 ◦ C and held there for 0.5 hrs. When the sample is heated the first time, the 

room temperature modulus increases monotonically with the maximum annealing temper- 

ature (Figure 2.11). After the first anneal and stabilization of the grain size and texture, 

additional thermal cycles do not enhance the modulus further, as shown in Figure 2.12. 

The biaxial modulus of either the as-deposited or annealed film appears to be the same 

whether it is being stretched or compressed (true at any temperature). The e-beam evapo- 

rated films show similar behavior. This result suggests that the deficit is not due to grain 

boundary cracking, which would predict a stiffer modulus in compression than in tension,


Table 2.5: Measured values of the biaxial modulus for films of thickness 1 µm or greater. 

Substrate Preparation Method Y (GPa) (as-dep.) Y (GPa) (ann.) 

Si(100) sputtered 197 ± 8 217 ± 8 

Si(100) e-beam evaporated 155 ± 8 158 ± 8 

Ge(111) sputtered 262 ± 8 - 

α-Al2O3(0001) sputtered 201 ± 8 225 ± 8 

fused silica sputtered 184 ± 8 - 

Figure 2.11: Biaxial modulus as a function of the maximum annealing temperature exposed 

toa1.88 µm thick copper film on Si(100).


Figure 2.12: Biaxial modulus as a function of the number of times thermally cycled to 

600 ◦ Cfora1.91 µm and 1.93 µm thick copper film. 

assuming the crack closes completely in compression. 

Biaxial Modulus as a Function of Temperature 

Figure 2.13 shows an example of how the biaxial modulus can be determined as a func- 

tion of temperature. As long as the elastic thermal cycle keeps a linear stress-temperature 

relationship, the biaxial modulus can be measured. 

Figures 2.14 and 2.15 depict the biaxial modulus as a function of temperature for the 

as-deposited films and annealed films, respectively. The moduli do not vary according to 

the temperature dependence of the elastic constants found in the literature [207] (indicated 

by the slope of the shaded areas), but drop off faster at higher temperatures. The shaded 

areas in Figures 2.14 and 2.15 represent the texture-dependent biaxial moduli calculated 

using equation (2.6). The width of the shaded areas is equal to twice the standard devi-


Figure 2.13: Stress as a function of temperature for a 1.19 µm thick Cu film on Si(100). 

The elastic thermal cycles for the measurements of biaxial modulus at five different temperatures 

are shown for the as-deposited film (a) and the thermally cycled film (b). 

ation of 22 GPa. As one can tell from Figure 2.13 (a), grain growth causes substantial 

stress relaxation once the temperature exceeds 150 ◦ C. Thus, the data in Figure 2.14 are 

marred by a change in microstructure and texture, which would raise the modulus at higher 

temperatures in the absence of a modulus deficit. 

Biaxial Modulus as a Function of Oscillation Frequency 

The effect of the heating/cooling rate (oscillation frequency) on the biaxial modulus was 

examined. A typical result is shown in Figure 2.16. The modulus seems to be unaffected 

by the oscillation frequency, at least within this frequency range.


Figure 2.14: Biaxial modulus as a function of temperature for as-deposited copper films. 

The error bar denotes two times the standard deviation. The shaded area represents the 

calculated biaxial moduli after accounting for texture. It is based on a value of 226±22 GPa 

at room temperature (cf. Table 2.4) and has the temperature dependence found in Simmons 

and Wang [207].


Figure 2.15: Biaxial modulus as a function of temperature for a copper film previously 

annealed at 600 ◦ C. The error bar denotes two times the standard deviation. The shaded 

area represents the calculated biaxial moduli after accounting for texture. It is based on 

a value of 256 ± 22 GPa at room temperature (cf. Table 2.4) and has the temperature 

dependence found in Simmons and Wang [207].


× 

× 

Figure 2.16: Biaxial modulus as a function of temperature for copper films previously 

thermally cycled to 600 ◦ C for different strain oscillation frequencies. 

Table 2.6: Comparison of the measured and calculated values of the biaxial modulus for 

the as-deposited films. 

Substrate Prep. Method Thickness Ymeas Ycalc Deficit 

(as-dep.) (as-dep.) 

Si(100) sputtered 1.25 µm 192 ± 8 GPa 226 ± 22 GPa 15% 


Si(100) e-beam evap. 0.94 µm 155 ± 8 GPa 203 ± 22 GPa 23% 

Ge(111) sputtered 1.87 µm 262 ± 8 GPa 253 ± 22 GPa ∼ 0% 

2.5 Discussion 

2.5.1 Texture 

The measured biaxial moduli are compared to the calculated moduli in Tables 2.6 and 

2.7. Excluding the Ge(111) sample, there still exists a modulus deficit, even when properly 

accounting for texture. The increase in modulus with annealing appears to be nearly fully 

accounted for by a sharpening of the 〈111〉 texture.


Table 2.7: Comparison of the measured and calculated values of the biaxial modulus for 

the annealed films. 

Substrate Prep. Method Thickness Ymeas Ycalc Deficit 

(ann.) (ann.) 

Si(100) sputtered 1.25 µm 224 ± 8 GPa 256 ± 22 GPa 12.5% 


2.5.2 Cause of the Modulus Deficit 

The dependence of the biaxial modulus on temperature suggests an anelastic effect. 

Grain boundary sliding is a likely contributor. Bohn and Schilling [97, 98, 17] have per- 

formed a series of internal friction experiments using both the vibrating reed and drumhead 

geometries to study grain boundary sliding in substrate-bonded and free-standing Al thin 

films approximately 4 µm thick. They have shown that the internal friction peak they stud- 

ied cannot be due to dislocation anelasticity, nor, for the substrate-bonded films, sliding 

of the grains at the substrate interface. Vinci et al. [237] have carried out frequency de- 

pendent tensile tests on free-standing Al and partly attributed the modulus deficit to grain 

boundary sliding. More recently, Kalkman et al. [122] have studied the Young modulus in 

free-standing Al, Au, and W films using a dynamic bulge test and found the modulus to 

decrease with oscillation frequency. 

However, the results in Tables 2.6 and 2.7 show that despite a ten-fold increase in grain 

size, the deficit is reduced only marginally. This also does not support the grain boundary 

cracking theory, since annealing would presumably heal the cracks. Furthermore, grain 

boundary sliding cannot explain the temperature dependence. The experimental frequen- 

cies are not within the range required to detect a temperature dependent difference in the 

biaxial modulus. 

According to the grain boundary sliding mechanism, the characteristic relaxation time


is dependent on the grain size and the temperature [122]: 

τ ∝ d n exp(Eact/kBT ) (2.7) 

where 1 < n < 2 and Eact is the grain boundary sliding activation energy (equivalent to 

the activation energy for grain boundary diffusion). The modulus in an internal friction 

experiment is given by [106] 

Y (ω, τ) ≈ YR 

� 

1 + δY 

YR 

τω 

π 

� � 

1 − exp − YU 

�� 

π 

YR τω 

(2.8) 

where YU is the unrelaxed modulus, YR is the relaxed modulus, δY is the difference be- 

tween the relaxed and unrelaxed modulus, and ω is the oscillation frequency. If we in- 

sert reasonable values for the relaxed and unrelaxed moduli into (2.8), for a frequency of 

10 −3 rad sec −1 and Eact ≈ 1.0 eV, the biaxial modulus at room temperature and the biaxial 

modulus at 300 ◦ C are nearly equal. The only remaining plausible anelasticity mechanism 

is dislocation anelasticity. 

A simple estimate of the reduction in modulus from dislocations is 

δY 

Y 

≈ 1 

20 ρℓ2 

(2.9) 

where ρ is the dislocation density and ℓ is the dislocation pinning length [68]. As shown 

by the schematic in Figure 2.17, it is possible for the total dislocation line length per unit 

volume, i.e., the dislocation density to remain the same at high temperatures while the 

pinning length increases by the thermal activation of dislocations past weak obstacles. This 

has some support in the literature [128, 48]. 

Although the frequency dependence of the modulus was not observed (Figure 2.16), 

the tested frequency range (8.73 × 10 −4 to 5.24 × 10 −3 rad sec −1 ) was not wide enough to 

expect any substantial changes in modulus.


l LT 

l HT 

Figure 2.17: Depiction of dislocation bowing at high temperatures. Thermal fluctuations 

help the dislocation to bow past weak obstacles, but the radius of curvature remains below 

that required for slip. 

The electron beam evaporated film has a larger modulus deficit compared to the sput- 

tered films; its average grain size is large, on the order of 1 µm [261], and the 〈111〉 texture 

is weak. Also, the “critical” temperature alluded to earlier is much lower for the evapo- 

rated film — about 165 ◦ C compared to about 300 ◦ C for the sputtered films. This may be 

related to the grain boundary mobility: unlike the sputtered films with trapped Ar atoms, 

the e-beam films have no impurities to inhibit grain growth. 

The e-beam microstructure should be contrasted with that of sputtered Cu on Ge(111); 

the grains extend from the substrate to the surface, but the in-plane grain size is approxi- 

mately 50 nm. Recall there is very sharp 〈111〉 texture for this film and no modulus deficit. 

Indeed, there may be some relationship between the width of 〈111〉 texture pole and the 

deficit, but more evidence is required for confirmation. If there is a relationship, this indi-


cates that the nature of the grain boundary is different in the different films. 

2.6 Conclusions and Future Research 

There are three main observations concerning these experiments. First, the biaxial mod- 

ulus of Cu films increases after thermal cycling due primarily to enhanced 〈111〉 texture. 

The deficit only changes marginally despite significant grain growth. Second, the modulus 

deficit varies from -23% to 0% depending on the deposition conditions and the substrate. 

Third, the magnitude of the deficit increases at higher temperatures. 

From the temperature dependence of the modulus deficit, it appears that it can be par- 

tially attributed to dislocation anelasticity. This has been reported for free-standing films 

in the literature [237], but not for substrate bonded films. It is unclear whether or not it 

is the dominant anelasticity mechanism, but it is anticipated that substrate-bonded films 

can experience more dislocation anelasticity than free-standing films due to the dislocation 

storage capacity of the former. 

More work should be carried out on the frequency dependence of the modulus; if the 

material studied is copper, this needs to be performed in conjunction with χ-scans or even 

ODF measurements. Both sputtered and evaporated films should be studied. 

It also would be interesting to repeat the biaxial modulus as a function of tempera- 

ture experiments on the cooling part of the thermal cycle rather than the heating part. As 

described in Chapter 3, the dislocation density and pinning length at a given temperature 

depend on whether or not the film is being heated or cooled. 

Finally, the Young modulus and Poisson ratio should be checked using the Y -cut quartz 

substrates both before and after annealing [262]. An examination of the changes in E and ν


due to annealing may further distinguish between the grain boundary cracking and anelastic 

models.

Chapter 3 

The Plastic Properties of Cu Thin Films 

The experiment described in this chapter is the search for a dislocation “boundary 

layer”; proof of the existence of such a layer would lend support to one of the major theo- 

ries on the strength of thin films: strain gradient plasticity. This introduction surveys some 

of the current theories regarding thin film plasticity. 

3.1 The Current Understanding of Plastic Properties of 

Thin Films 

Perhaps the first to carry out a systematic investigation of plastic properties in thin films 

was Murakami at IBM [166, 161, 160, 162, 165, 167], who studied thermal strains in Pb 

thin films for the expressed purpose of developing reliable Josephson junctions [168, 163, 

164]. His results stimulated two of his colleagues, Chaudhari and Ronay, to propose the 

first quantitative models of plastic flow in thin films in 1979 [34, 37, 35, 36, 191]. In the 

late 1980s, Nix and his group at Stanford began studying thin film stresses by performing 

49

Chapter 3: The Plastic Properties of Cu Thin Films 50 

wafer curvature measurements during thermal cycling [62]. Aluminum and aluminum alloy 

thin films were extensively studied at first because of their use as interconnect materials 

[53, 233, 231, 232]. As it became apparent that copper would be more cost-effective than 

aluminum (having a lower resistivity and a higher melting temperature), researchers at 

Stanford [263, 238, 236] and at the Max Planck Institute for Metals Research in Stuttgart 

[124, 125, 10] began to focus on copper thin films. 

The “classic” reviews in thin film mechanical properties are by Nix and Arzt [54, 172, 

173, 6], although some of the more recent papers help to clarify several issues as well as 

raise more questions. In this introduction, I will focus on the models currently used to 

describe low temperature flow, but first I will present a brief description of deformation 

mechanisms at high temperatures (above ∼ 200 ◦ C) since they have some bearing on the 

room temperature yield strength. The discussion is restricted to unpassivated thin films. 

3.1.1 High Temperature Deformation 

The earliest model of high temperature deformation allowed for grain sliding at the 

film/substrate interface [221, 223, 222]; for films in which the grain size is less than the 

film thickness, the data agreed with the model. If, however, the adhesion between the film 

and substrate is strong enough to prohibit diffusion along the interface, only partial stress 

relaxation by grain boundary diffusion can take place. This supposition was first made 

by Jackson and Li [110], who predicted the creation of strain gradients in the film due 

to this phenomenon. More recently, Gao et al. [73] have made a mathematical model to 

predict the decay of normal tractions at the grain boundaries with time. They also predicted 

that so-called grain boundary diffusion wedges could serve as sources of dislocations. For


copper films grown and annealed in high vacuum (HV), however, surface diffusion seems 

to be shut off entirely (perhaps due to a metastable oxide), leading to negligible stress 

relaxation by grain boundary diffusion, for which surface diffusion is a required pathway. 

The predominant stress relaxation mechanism in HV-grown Cu at high temperatures is 

power-law creep [125], or at least the mathematical formulation of power-law creep seems 

to fit the data. The actual mechanism by which power-law creep works in thin films is not 

yet understood. Incidentally, it is important to distinguish between experiments performed 

in clean and “ultra-clean” conditions. For example, UHV-grown Cu films annealed in a 

UHV furnace do experience substantial surface diffusion [246]. 

3.1.2 Low Temperature Deformation 

At low temperatures plasticity operates by dislocation motion. Most dislocation models 

have not been successful in simulating the low temperature portion of the stress-temperature 

profile, although work on this is apparently in progress [246]. Instead, most models focus 

on a prediction of the room temperature yield strength. 

At room temperature, the yield strength hierarchy is as follows: 

σy, passivated >σy, unpassivated >σy, free−standing >σy, bulk, (3.1) 

where σy is the yield strength. For free-standing thin films and the bulk, the yield strength 

can be determined by the usual 0.2% plastic strain criterion. For unpassivated and pas- 

sivated thin films (the use of these terms imply the films are substrate-bonded), the yield 

strength is assumed to be the flow stress at room temperature as determined by wafer curva- 

ture measurements [53], although this assumption is not always valid. Thus, from here on, I


will call what others label the yield strength, σy, i.e., the biaxial stress at room temperature 

after a thermal cycle, the flow strength, σflow, instead. 

The flow strength has several origins, each of which depends on the nature of the resis- 

tance to dislocation motion. There are three main contributions to the flow strength of thin 

films: the constraint of the film by the substrate, the small grain size, and work hardening. 

The mathematical form of these models can be written as 

σconstraint = A 

, (3.2) 

σ I grain size 

or σ II 

grain size 

tf 

B1 = d , (3.3a) 

= B2 

√d , (3.3b) 

and σhardening = C √ ρ (3.4) 

where tf is the film thickness, d is the grain size, ρ is the dislocation density, and A, Bi, 

and C are model-dependent parameters that are discussed below. As we will see, these 

equations are only approximate. Researchers have usually taken the total flow strength to 

be equal to the superposition of all these strengthening mechanisms (dislocation obstacles) 

[124, 10]: 

σflow = σconstraint + σgrain size + σhardening, (3.5) 

although this is true only if every dislocation must overcome all the obstacles simultane- 

ously. Equations (3.3b) and (3.4) were previously derived for bulk materials but remain 

valid for thin films; the main difference is that the parameters B2 and C are much larger for 

thin films [124].


There is some experimental verification for equation (3.5): Venkatraman and Bravman 

[233] have shown that for aluminum thin films, 

σflow = σconstraint + σgrain size 

(3.6) 

where the grain size refers to the in-plane size. However, the work hardening contribution 

is ignored (perhaps for good reason, as discussed below), and there remains some debate 

on whether equation (3.3a) or (3.3b) is more appropriate [124, 56] for the grain size rela- 

tionship. 

Many competing theories are developed for the empirical models described above, but 

often none of them can be distinguished experimentally due to the difficulty in seeing how 

an ensemble of dislocations behaves in situ. The situation is naturally worse for poly- 

crystals than for single crystals, yet there still remains contentious debate on the nature of 

work hardening in single crystals [134]. A natural approach nowadays is to perform com- 

puter simulations of dislocation interactions [171], but we remain a long way away from 

simulating realistic structures. The field is too young for the development of a “statistical 

mechanics” of dislocations. 

In a recent review, Gao et al. [71] categorize the current theories according to the length 

scale in which they are operative. For deformation length scales larger than 100 µm, clas- 

sical plasticity works. Strain gradient plasticity is applicable in the 0.1 µm − 10 µm range, 

and discrete dislocation models work in the < 0.1 µm regime. Thus, strain gradient plas- 

ticity can successfully fill the void left by the lack of a statistical mechanics formulation. 

I will begin by discussing the discrete dislocation models and how they relate to the 

strength of thin films. Although Gao et al. [71] have stated strain gradient plasticity to 

be the theory of choice in the 1 µm regime, nearly all of the models in the literature use


discrete dislocation models to predict the flow strength for films of this thickness. 

Discrete Dislocation Models 

THE CONSTRAINT OF THE SUBSTRATE 

One of the more famous “criteria” in thin film physics is the Matthews-Blakeslee cri- 

terion, which gives the maximum film thickness of an epitaxial layer before nucleation of 

a periodic array of misfit dislocations [152, 153, 151]. They arrived at their criterion by 

considering the forces necessary to propagate a threading dislocation across an epitaxial 

film. Freund later generalized and refined their result using a dislocation dynamics ap- 

proach [65]. Here, I use Nix’s interpretation of Freund’s work [172], which includes many 

simplifications, rendering it nearly equivalent to the Matthews-Blakeslee result. The work 

per unit length performed by the coherency stress during misfit dislocation formation is 

Wlayer = 

cos λ cos φ 

σfbtf, (3.7) 

sin φ 

where b is the Burgers vector, σf is the coherency stress, φ is the angle between the glide 

plane normal and the film normal, and λ is the angle between the Burgers vector and the 

film normal (see Figure 3.1); the geometric term cos λ cos φ is the Schmid factor. The work 

to make a unit length of edge dislocation at the interface is (after the image forces at the 

interface of a stiff substrate are considered [96] and ν = νs = νf is assumed) 

Wdis = 

b2 2GfGs 

4π (1 − ν) (Gf + Gs) ln 

� � 

βtf 

b 

(3.8) 

where Gi are the shear moduli of the film and substrate, β is a dislocation core radius cutoff 

(≈ 2), and elastic isotropy in the film and substrate is assumed. By equating (3.7) to (3.8),


λ φ 

Figure 3.1: Diagram showing the nucleation and propagation of dislocations along a slip 

plane from the surface to the grain boundaries and film/substrate interface. 

we arrive at 

tf,c = 

sin φ 

cos λ cos φ 

from which the critical thickness can be determined. 

2GfGs b ln 

(Gf + Gs) 

� βtf,c/b � 

4π (1 − ν) σf 

(3.9) 

Nix [172] recognized the importance of this equation. If the thickness of the film is 

larger than the critical thickness, σf is equal to the stress required to move a dislocation 

along the interface. If this is the limiting factor for dislocation motion, 

σconstraint = 

sin φ b 

� 

GfGs 

cos φ cos λ 2π(1 − ν)tf 

Gf + Gs 

ln 

� �� 

βtf 

. (3.10) 

b 

As described below, Chaudhari was actually the first to derive this relationship [36]. This


model predicts the experimentally observed 1/tf dependence and is often applied to poly- 

crystalline films. 

Baker et al. [9, 181] have used this model to explain the Bauschinger effect (σy(tension) 

>σy(compression)) often found in stress-temperature profiles. They believe that when 

lowering the stress from room temperature by heating, the dislocations run backwards eas- 

ily, removing the energetically expensive line segments from the film/substrate interface. 

Thus, the plasticity found in compression is driven both by an applied compressive stress 

and the removal of misfit dislocation line length. 

Weihnacht and Brückner [242, 243, 245] have an alternate explanation for the Bauschinger 

effect: pile-ups. They assume that dislocations nucleate at the surface and pile up at the 

substrate interface when cooling from the annealing temperature. The same dislocations 

then undergo “easy” back gliding (aided by the mutually repulsive forces within the pile- 

up) across the same slip planes when the film is heated (Figure 3.2). This back gliding can 

occur even when the stress state is still positive, i.e., in tension. 

Weihnacht and Brückner’s model has some experimental support: using in situ thermal 

cycling of Al/Si(100) in the TEM, Allen et al. [1] showed that dislocations move in and out 

along the same slip planes as the temperature is cycled. 

THE GRAIN SIZE EFFECT 

As mentioned above, Chaudhari [36] was the first to derive (3.10). Rather than using 

a dislocation dynamics method, he relied on an energy minimization approach, calculating 

the total strain and dislocation energy in the film and setting the derivative with respect to 

strain equal to zero. He obtained the same equation as above, but with ln(2tf/b) + 1 rather


σ 

Figure 3.2: Schematic showing the predicted dislocation behavior at different points on the 

stress-temperature profile. Based on the diagram by Weihnacht and Brückner [245]. 

than ln βtf/b. In addition, Chaudhari considered the strengthening due to grain boundaries: 

σ I grain size = 

� � 

sin φ Gfb 

1.87 + 

cos φ cos λ 4πd 

1.29 

�� 

d 

ln + 1 , (3.11) 

1 − ν b 

as well as strengthening due to ledge formation at the film surface: 

σledge = 

sin φ γ 

cos φ cos λ tf 

T 

(3.12) 

where the grain size d is the cut-off radius for the dislocations near the grain boundary, and 

γ is the surface energy per unit area. The numerical values in (3.11) are due to some fine 

details and are not important here. More recently, Thompson [218] independently derived 

a similar equation. 

In bulk materials, Hall [92] and Petch [185] found the flow strength to scale with the


inverse square root of grain size: 

σ II 

grain size = σ0 + kHPd −1/2 

(3.13) 

where σ0 and kHP are material-dependent quantities. This empirical equation has been 

justified several ways [57, 143, 8]. The original explanation was that the flow strength is 

equivalent to the critical stress (caused by dislocation pile-up) in a grain needed to activate 

slip in the neighboring grain. The value of kHP is larger in thin films than in the bulk [124]. 

WORK HARDENING 

Taylor [216, 217] was the first to derive 

σwork hardening = kTbGρ 1/2 , (3.14) 

and numerous subsequent derivations can be found [170]. According to the Taylor deriva- 

tion, the stress field a distance r away from a screw dislocation is given by bG/2πr. If 

we presume slip occurs at random points in the crystal by the separation of positive and 

negative dislocations, and on the average the dislocations move a distance ℓ before be- 

ing blocked, the plastic shear strain equals ρbℓ, which leads to the Taylor relation with 

kT = 1/2π. Other derivations, including those that consider image forces on moving 

dislocations (long-range interactions) and crossing dislocations on different slip systems 

(short-range interactions) can increase the value of kT to 1 [191]. Another way of deriving 

(3.14) is simply to insert the relationship ℓ = ρ −1/2 into the Orowan equation (1.11). More 

recently, Weihnacht and Brückner [244] have developed a work hardening model that con- 

siders the interactions of a threading dislocation with a parallel array of misfit dislocations. 

Note that an implicit assumption in most dislocation models is that the flow is in a 

drag-controlled regime rather than an obstacle-controlled glide regime. New observations


by Kobrinsky and Thompson [128] and Dehm et al. [48] have shown very short mean 

free paths for dislocation motion: 80 nm for annealed Ag and 70 nm for annealed Cu. 

These lengths were much smaller than the grain size. The dislocations moved in a “jerky” 

fashion characteristic of dislocations requiring thermal fluctuations to overcome obstacles 

(forest dislocations) [130]. They did not observe dislocation sweeps across the slip plane 

as depicted in Figure 3.1. These observations suggest more work needs to be done to 

understand the kinetics of slip in thin films; also, the work hardening contribution to the 

flow strength may be the most important one. 

Using these discrete dislocation models and the superposition equation (3.5) was the 

first step in trying to understand the flow strength in thin films. In general, equation (3.5) 

tends to underestimate the experimental thin film flow strength. Any good agreement be- 

tween the two should be considered fortuitous given the large number of assumptions in- 

volved. Moreover, equations (3.3a) (or (3.3b)) and (3.4) cannot be independent; Li has 

suggested the equations are in fact redundant since ρ tends to scale with 1/d [142]. 

Strain Gradient Plasticity in Thin Films 

The starting point for any model predicting flow strength in substrate-bonded films is 

the effect of the substrate on dislocation behavior since the mechanical properties of free- 

standing films can be predicted using the Hall-Petch relation. According to the theory of 

Head [96], dislocations should be repelled by the rigid substrate if the shear modulus of 

the substrate is larger than that of the film. Weihnacht and Brückner [242] have indirect 

TEM evidence of such pile-ups for Cu thin films on oxidized Si substrates. Dehm and 

Arzt have supplied somewhat contradictory evidence in that they observe dislocations in


Cu that are attracted and pulled into an amorphous silicon nitride interface [47]. This only 

happens after a some time under the electron beam, however, and may be due to core 

spreading caused by irradiation-induced diffusion [72]. For this discussion, I assume the 

rigid substrate is an effective barrier to dislocation propagation. 

The presence of a dislocation barrier creates gradients of plastic strain since dislocations 

pile-up at the interface and are emitted from the surface. As discussed in Chapter 1, plastic 

strain gradients imply the presence of geometrically necessary dislocations (GNDs) for 

compatibility. 

Ronay [191] was the first to use strain gradient plasticity to explain the large thin film 

flow strength. Her model will be discussed in some detail because it is easy to compare her 

SGP formulation to the discrete dislocation models; in fact, her formulation is akin to using 

SGP to derive similar relationships. She assumed that the flow strength could be expressed 

as the superposition of statistically stored dislocations (SSDs), and long-range (LR) and 

short-range (SR) interaction GNDs: 

σflow = σSSD + σLR, GND + σSR, GND. (3.15) 

For the long-range interactions, she based her results on Ashby’s treatment of rigid plate- 

like particles in a softer matrix [7]. If a shear strain ɛsh is imposed on the film, the amount 

of shear at the surface is equal to ɛsh, while the shear at the interface is 0. Thus, the 

average gradient on a single slip system is ɛshζ/t if the grain extends from the surface to 

the interface. Here, ζ is a geometric factor that depends on the grain orientation and the 

slip plane (= sin 70.5 ◦ for FCC materials). Thus, according to equation (1.13) the density


of GNDs is given by 

ρG = ɛsh 

. (3.16) 

bζ tf 

Using the Taylor relation (3.14), with kT = 1/π (the estimate for screw dislocations near 

an interface), 

σLR, GND = 1 

π Gf 

� bɛsh 

Ronay then considered grain boundary strengthening and found 

ζ tf 

� 1/2 

. (3.17) 

ρG = 2ɛsh 

. (3.18) 

bd 

This is the two-dimensional counterpart to Ashby’s three-dimensional result (4ɛsh/bd). 

Since the deformation of a grain bonded to a substrate requires multiple slip for compati- 

bility, this is a short-range interaction. As described above, the Taylor relation can be used 

with kT = 1: 

σSR, GND = Gf 

� 2bɛsh 

This is for a single grain. The stress for a polycrystal is 

σSR, GND = mGf 

where m is the Taylor orientation factor. 

Thus, the flow strength according to Ronay is 

σRonay = σSSD + 1 

π Gf 

� bɛsh 

ζ tf 

d 

� 2bɛsh 

� 1/2 

d 

� 1/2 

. (3.19) 

� 1/2 

+ mGf 

� 2bɛsh 

d 

(3.20) 

� 1/2 

. (3.21) 

This differs from the above models in that it does not have the inverse film thickness depen- 

dence. As Ronay mentions, both the second and third terms on the RHS of this equation 

arise from the rigid substrate constraint.


Hutchinson [109] has recently used the continuum higher order strain gradient plasticity 

theory (SGP) to derive the inverse thickness dependence of the flow stress. His derivation 

explicitly distinguishes between the flow stress and the yield strength in a material. In the 

case of a thin film bonded to a rigid substrate there are only stretch gradients, with an as- 

sociated length scale, ℓSG, and no rotation gradients. Fleck and Hutchinson’s formulation 

[61] also considered SSDs as well as GNDs, which enabled Hutchinson to determine the 

strength as a function of plastic strain, degree of plastic constraint, and length parameter, 

ℓSG. He found that the flow stress can rise to over three times the yield strength at approxi- 

mately 1% strain for a film 2 × ℓSG thick. A major distinguishing feature of this theory is 

the prediction of a dislocation boundary layer at the interface, with a thickness on the order 

of the average distance of dislocation travel. 

There are several hints in the literature that strain gradients exist in thin films, but no 

boundary layer has ever been observed. Murakami demonstrated the existence of strain 

gradients in planes parallel to the surface for 1 µm thick Pb/Si(111) films cooled to 77 K 

[160]. The strain profile was determined by analyzing the asymmetry of the (333) peak 

(taken in the powder diffraction geometry) with Houska’s method [103]. His measurement 

was performed in situ, with the sample mounted on a liquid helium cooled diffraction stage. 

The data were fit with the empirical equation 

ɛ⊥(z) = ɛint exp � −(tf − z)/z ∗� 

(3.22) 

where ɛ⊥ is the elastic strain along the surface normal, ɛint is the strain at the interface, and 

z ∗ is a characteristic length. Murakami found a strain of 0.18% at the interface and 0.04% 

at the surface. The main problem with this measurement is that the strains parallel to the 

interface (the biaxial strains, ɛ�) cannot be measured directly. Doerner and Brennan [52]


have also observed strain gradients in annealed Al thin films (260 nm and 600 nm thick) on 

Si(100) using grazing incidence diffraction (GID), which does allow a direct measurement 

of ɛ�. The strain rose from near zero at the surface, and reached a plateau 5 nm into the 

260 nm thick film and 200 nm into the 600 nm thick film. 

Most recently, Baker et al. [10] have found that for Cu thin films bonded to Si sub- 

strates, the dislocation densities are temperature-dependent and strain-independent. This 

accurately describes the behavior of geometrically necessary dislocations [191]. 

3.2 Determining the Strain Profile 

The complete plastic strain profile has never been determined experimentally. If the 

exact distribution can be measured, and in particular, the presence of a boundary layer 

near the substrate interface proven, this would allow for an estimate of ℓSG and a better 

understanding of the configuration of GNDs. This is the goal of our experiment. 

For a thermally cycled film, the total biaxial strain, ɛ total 

� , is approximately known since 

it can be calculated from �α�T , where �α is the difference in CTE between the film and 

substrate (discussed in Chapter 2). Since x-ray diffraction can measure the elastic strain as 

a function of penetration depth, ɛ�(z), the plastic strain profile can be determined using the 

relationship 

ɛ pl 

� (z) = ɛtotal � − ɛ�(z). (3.23) 

Although our goal is to determine ɛ pl 

� (z), I will more often refer to ɛ�(z) since this is 

the measured parameter. Complicating matters is that our interest is specifically in the 

plastic strain from dislocation flow; where this begins on the stress-temperature profile


is unclear. Another complication is that it is possible for strain gradients to arise from 

diffusive processes rather than dislocation processes [73] as well as from grain boundary 

sliding [17]. 

One previous attempt at measuring strain gradients due to SGP was made by Leung 

et al. [141]. They studied thermal strains in gold thin films of various thickness using the 

scattering vector method described by Genzel [79, 80, 77]. An advantage of the scattering 

vector method is that one can measure the strain as a function of penetration depth for a set 

of grains with the same orientation with respect to the surface normal (grain population). 

Leung et al. did not find strain gradients, but their data contained large errors caused by 

reliance on a low diffraction angle; in addition, no error bars were included, so small strain 

gradients could not be discerned. Furthermore, the strains measured were not parallel to 

the surface. 

We attempted to probe the elastic strain distribution, ɛ�(z), using the grazing incidence 

diffraction method [52]. This allows an investigation of the surface strains when the x-rays 

enter the sample from the surface side and an investigation of the interfacial strains when 

the enter the sample from the substrate side. 



Most of the Cu films were grown on oxidized Si(100) substrates approximately 400 µm 

thick. For better sensitivity with the curvature apparatus, the thinner films were deposited 

onto 97 µm thick bare Si(100) substrates. Before deposition, the baseline curvatures of the


substrates were measured. 

Sputtered Cu films (from a 99.99 at.% Cu target) of thicknesses 0.22 µm, 0.5 µm, 

0.87 µm, 1.25 µm, and 1.73 µm were deposited with an Ar pressure of 1.1 × 10 −2 Pa 

after reaching a base pressure of 1.3 × 10 −5 Pa. The sputtering chamber is described by 

Spaepen et al. [213] and shown in Figure 1.7. The sputtering parameters are given in Table 

1.2. 

For the films deposited onto bare Si(100) substrates, a thin amorphous silicon nitride 

(a-SiNx) layer, nominally 40 nm thick, was put down before depositing the Cu to act as 

a diffusion barrier between the film and the substrate. This layer was also deposited on 

the backside of the substrate after the copper deposition. The sputtering parameters for 

the nitride deposition are in Table 2.2. Prior to deposition, the substrates were sputter- 

cleaned for 2-4 minutes. The substrates were also water-cooled from the backside, leading 

to temperature rise of 10 ◦ C after three hours of deposition. A typical deposition rate was 

0.15 nm/s. 


After deposition, the thicknesses of the films were measured with a Tencor Alpha-Step 

200 Profilometer. For the thinner films (less than one micrometer thick), the thickness was 

also measured with Rutherford Backscattering Spectrometry (RBS). 

Microstructural characterization was performed by plan-view and cross sectional trans- 

mission electron microscopy (TEM). The film purity was checked by RBS and energy 

dispersive spectroscopy (EDS) in the TEM.



The wafer curvature apparatus was described earlier in Chapter 1, and the curvatures 

were converted to stresses using the Stoney equation (1.16). The thermomechanical proper- 

ties of the Si(100) substrates are listed in Table 2.3. Thermal cycles to 660 ◦ C were carried 

out for one film of each thickness at rates of 5 ◦ C/min. Another set of films of the same 

thicknesses was left in the as-deposited state. 


The GID measurements were carried out at Beamline X22C at the National Synchrotron 

Light Source at Brookhaven National Laboratory. The GID geometry is shown in Figure 

1.6 (c). A0.1 ◦ soller slit was placed before the detector to ensure a small diffraction peak 

width. The energy of the incoming photons was tuned to 8960 eV, which is just below the 

Cu K edge. For studies in which the x-rays entered through the back of the substrate, the 

third order reflection of the Si(111) monochromator was used so that the incoming energy 

was 26.9 keV. The energy window of the scintillation detector was adjusted accordingly. 

In a GID measurement, the diffraction angle 2θ does not lie in a plane parallel to the 

incoming beam so a correction must be made to convert the diffractometer-specified 2θ ′ to 

the true 2θ. This geometry is displayed in Figure 3.3, and the conversion is given by 

cos 2θ = cos 2α cos 2θ ′ 

(3.24) 

where α is the incoming angle and 2θ ′ is the detector motor position. The correction was 

checked using a CeO2 powder standard from NIST (SRM 674a).


2θ 

2θ 

2α 

Figure 3.3: The scattering geometry for the GID measurements. As shown in (a), the 2θ 

value is not the same as the motor position for the detector, 2θ ′ . The correction can be 

made using the well known formulas for the right-spherical triangle shown in (b). 

where 

The (1/e) penetration depth, τ, can be calculated using the following equation [52]: 

2q 2 = 

τ = λ 

4πq 

2θ 

2α 

2θ 

(3.25) 

� 

2δ − α 2� 

�� + α 2 �2 − 2δ + 4β 2 

�1/2 . (3.26) 

Here, λ is the x-ray wavelength, q is the scattering vector, α is the incidence angle, and δ 

and β are associated with the real and imaginary parts of the index of refraction, n, i.e., 

n = 1 − δ − iβ (3.27) 

and may be looked up in any x-ray reference table (e.g., www-cxro.lbl.gov). 

Following Leung et al. [141], we only searched for strain gradients in the three thickest 

films, (0.87 µm, 1.25 µm, and 1.73 µm) since a strain gradient from diffusional flow is ex- 

pected for the thinner films. The strain profiles in the thermally cycled films were compared


to those in the as-deposited films. The measurements were made using the (220), (200), 

and (111) diffraction peaks with the incidence angle varying from 0.3 ◦ to 4.3 ◦ . Since the 

texture poles shown in Chapter 2 are broad, a tilt of 4.3 ◦ does not significantly vary the 

sampled grain population. 

For two of the samples (the as-deposited and annealed 1.25 µm thick Cu), the strains 

measured by the synchrotron were checked with sin 2 ψ measurements taken on the diffrac- 

tometer at Harvard University (Figure 1.5). The Kamminga method and the crystallite 

group method (Appendix A) were used for the as-deposited and annealed films, respec- 

tively. 

3.4 Results 


Cross-sectional views of an as-deposited and an annealed 0.87 µm thick copper film are 

shown in Figure 3.4. The plan-view micrographs were presented in the results section of 

Chapter 2. The as-deposited films appear to have a thin columnar morphology with an in- 

plane grain size of about 50 nm, while the annealed films have graded morphology with a 

grain size of about 500 nm near the surface of the film and a bimodal grain distribution near 

the substrate-side of the film, with an average grain size of 200 nm. Even after annealing, 

the grains do not extend from the surface to the substrate. 

copper. 

As discussed in Chapter 2, both RBS and EDS show no impurities in the bulk of the


Figure 3.4: Cross-sectional views of an as-deposited (a) and annealed (b) 0.87 µm thick 

copper film. 

(a) 

(b)


Figure 3.5: Deposition stress for ion beam sputtered Cu films as a function of total film 

thickness. 

3.4.2 Curvature Results 

The deposition stresses for films of various thickness are shown in Figure 3.5. A caveat 

should be mentioned here: the filmsof0.22 µm and 0.50 µm thickness have a silicon 

nitride layer on the backside of the substrate to cancel out the curvature change due to the 

silicon nitride layer in between the substrate and film. If the nitride thicknesses on either 

side of the Si substrate are not equal, this could result in an error in the deposition stress 

measurement. The thickness error is expected to be small, however. 

Figure 3.6 shows the room temperature flow strength after thermal cycling for the cop- 

per films. The inverse thickness relationship predicted by σconstraint and the higher order 

SGP theory appears to hold. 

The three thermally cycled samples tested for strain gradients have the stress-temperature 

profiles displayed in Figure 3.7.


Figure 3.6: Room temperature flow strength as a function of inverse thickness for Cu samples 

thermally cycled to 660 ◦ C. 

Figure 3.7: The stress-temperature profiles for the Cu films used in the strain gradient 

studies.


d hkl (h 2 +k 2 +l 2 ) 1/2 [Å] 

5.420 

5.415 

5.410 

5.405 

5.400 

0 

1 

2 

α [degrees] 

Figure 3.8: Lattice parameter of CeO2 as a function of incidence angle, α. The lattice 

parameters were measured with the (331) reflection. 

3.4.3 X-Ray Diffraction Results 

As mentioned earlier, a standard reference material, CeO2, was used to calibrate the 

instrument and check the 2θ correction (3.24). Figure 3.8 shows lattice parameter of CeO2 

as a function of incidence angle after the correction is applied. The measurement was 

carried out using the (331) reflection. The large error bars stem from poor intensities 

caused by nonuniform spreading of the CeO2 powder on a substrate. That the powder 

shows no strain gradients gives us confidence in our measurements. 

Measurements from the Surface of the Film 

The strain distributions probed by x-rays entering the surface of the Cu are shown in 

Figure 3.9. The solid and dotted curves refer to the thermally cycled films and as-deposited 

films, respectively, and the reflection plane used is indicated in the legend. The vertical 

3 

4


Figure 3.9: Elastic strain gradients in Cu films. The solid and dotted curves refer to the 

thermally cycled films and as-deposited films, respectively. The reflection is denoted by 

different markers, as shown in the legend. The plastic strain is given by 0.9% minus the 

elastic strain. 

dashed lines indicate the points at which ∼ 63% of the diffracted intensity comes from 

the total thickness of the film. To the right of that line, more of the diffracted intensity 

originates from the film/substrate interface. If we take the maximum annealing temperature 

(660 ◦ C) for calculating the thermal strain, the total strain in all the films is ≈ 0.9%, and 

the plastic strains can be determined with equation (3.23). 

Measurements from the Film/Substrate Interface 

Despite the high brilliance of the synchrotron, we were unable to obtain a diffraction 

peak from the Cu when the x-rays entered through the back of the substrate. After scattering 

from the Cu, nearly all the photons were absorbed by the Si.


Figure 3.10: sin 2 ψ-plot for an as-deposited Cu film. 

Comparison with sin2ψ 2ψ 2ψ Measurements 

Figures 3.10 and 3.11 show the sin 2 ψ plots for the as-deposited and thermally cycled 

1.25 µm thick films, respectively. The reference lattice parameter was taken to be the lit- 

erature value, aref = a0 = 0.3615 nm [115]. This agreed well with strain-free lattice 

parameter determined by sin 2 ψ analysis (Appendix A). The strains agree with those from 

the GID experiments, which gives us confidence in the absolute value of the lattice parame- 

ters. There is a significant difference between the stresses in the 〈111〉 grains and the 〈100〉 

grains for the thermally cycled film. The stresses in the randomly oriented grains fell along 

the 〈111〉 sin 2 ψ line. 

ψ 

σ


3.5 Discussion 

σ 

Figure 3.11: sin 2 ψ-plot for a thermally cycled Cu film. 

σ 

The cross-sectional TEM micrographs show grain boundaries and twins parallel to the 

surface, which makes it difficult to analyze the strain distribution in terms of an inhomoge- 

neous dislocation distribution. Nevertheless, the films exhibit extremely high flow stresses, 

as depicted in Figure 3.7. If the large strength is primarily due to strain gradient plasticity, 

the strain gradients should be measurable even though the nature of the boundary layer may 

be different. 

According to the texture results in Chapter 2, the texture of the thermally cycled films is 

predominantly 〈111〉; thus, the stress in the 〈111〉-oriented grains, 489 MPa (Figure 3.11), 

is taken to be the overall biaxial stress, and the stress within the 〈100〉 grains is ignored. 

With this value for the annealed film and −124 MPa for the as-deposited film, the x-ray 

measurements underestimate the stress measured by substrate curvature in Figure 3.7 by 

ψ


about 15-20%, which is insignificant given the errors in both measurements. The sin 2 ψ 

results films are in agreement with the synchrotron measured strains, however. The large 

difference in stress between the 〈111〉 and 〈100〉 oriented grains is consistent with the re- 

sults of Baker et al. [10]. 

Due to absorption, the biaxial strain as a function of (1/e) penetration depth, ɛ�(τ), is 

related to the strain as a function of position, ɛ�(z),byafinite thickness Laplace transform 

[13]: 

ɛ�(τ) = 

� tf 

0 dzɛ�(z) exp (−z/τ) 

. (3.28) 

dz exp (−z/τ) 

� tf 

0 

Figure 3.12 shows a hypothetical strain profile that demonstrates relaxation at the surface 

and a boundary layer near the interface (the uppermost solid curve). The finite thickness 

Laplace transform of such a profile is shown as the closed circles in the figure. The shape of 

the finite thickness Laplace transform is invariably the same no matter what the magnitude 

of the boundary layer (e.g., from 0 to a full 0.9% strain at the interface). Only the surface 

relaxation can be detected from ɛ�(τ). Although in theory ɛ�(z) can be retrieved using the 

inverse Laplace transform, the inverse problem is nearly impossible given the error bars in 

the data; even with infinitesimal error, the unstable nature of the inverse problem would 

makeitdifficult to identify a boundary layer. 

This emphasizes the necessity of making lattice parameter measurements from the back 

of the substrate. This is a difficult technique that has not even been attempted before accord- 

ing to the literature (and now we know why). The main problem is that the measurement 

is performed at grazing angles; this requires the x-rays to enter and scattered x-rays to 

exit through the entire width of the substrate (≈ 0.25 in). The closest measurement to this 

that has been successful is a reflectivity measurement from an interface, which researchers


Biaxial Elastic Strain [%] 

0.6 

0.4 

0.2 

0.0 

-0.2 

0.0 

0.5 

ε 

ε τ 

1.0 

Depth [µm] 

Figure 3.12: Hypothetical strain profiles and their normalized finite thickness Laplace 

transforms for a 1.8 µm thick Cu film. 

performed to determine the nature of the solid/liquid interface for liquid Pb on Si [189]. 

Thus, for now, I will focus only on the surface strain behavior in Figure 3.9. 

3.5.1 The As-Deposited Films 

Intrinsic stresses in as-deposited films have been the subject of several theses ([204, 

186, 49]) and reviews ([54, 254, 129, 212]). Only the relevant points are discussed here. 

Tensile stresses in as-deposited films originate from excess volume. For the microstruc- 

ture seen in Figure 3.4, one can see the main source of excess volume: grain boundaries. In 

addition, there is a contribution from vacancy annihilation. Tensile stresses are very com- 

mon in high-vacuum deposited films, particularly those grown by evaporation. The stresses 

are often less tensile in UHV-grown films, however, presumably due to enhanced adatom 

mobility. 

ε τ 

ε 

1.5


The usual explanation for compressive stresses in ion beam sputtered films is that neu- 

tral Ar atoms are back-reflected from the target toward the growing film [50, 51, 253]. The 

energies of these back-reflected atoms are in the hundreds of eV range, while the energies 

of the deposited atoms are in the 10 − 40 eV range. If the Ar gas pressure is low enough, 

the back-reflected atoms can arrive at the film with little energy loss, and the momentum 

transfer can produce a compressive component on top of the tensile component. 

The Strain Profile 

The surfaces of the as-deposited films are in tension (on the order of 200 MPa) although 

the average stress in the film is compressive (on the order of −150 MPa). Although the 

compressive stress component is not obvious from the ɛ�(τ) profile in Figure 3.9, the hy- 

pothetical ɛ�(z) profile (the lower solid line in Figure 3.12) shows how it can be obscured. 

It is not currently understood how this profile develops, and further work is required. 

Anisotropy 

From our results in Chapter 2, we know the biaxial modulus varies with orientation in 

the following way: 

Y〈111〉 > Yrandom > Y〈100〉. (3.29) 

Thus, if the stress state is the same in all grains (the isostress limit [10]), then 

ɛ〈100〉 >ɛrandom >ɛ〈111〉. (3.30) 

During a GID measurement, the scattering vector is nearly parallel to the plane of the film. 

The (220) reflection contains the 〈111〉 directions, the (200) reflection contains the 〈100〉


directions, and the (111) reflection contains the 〈110〉 directions. According to our texture 

results in Chapter 2, the as-deposited films are predominantly 〈111〉 textured but also have 

〈100〉 and random components. Thus, the (220) strains represent the 〈111〉 textured grains, 

the (200) strains represent the 〈100〉 textured grains, and since there is no 〈110〉 texture, the 

(111) strains may represent the randomly oriented grains. Then, according to Figure 3.9, 

equation (3.30) holds. 

3.5.2 The Thermally Cycled Films 

The Strain Profile 

The thermally cycled films all show strong relaxation at the surface. (This qualitatively 

agrees with what happens if the thermally cycled film delaminates: it immediately curls 

with negative curvature). The relaxed “layer” is approximately 300 nm thick for all the 

thermally cycled films. The cross-sectional micrograph (Figure 3.4 (b)) suggests that sur- 

face diffusion allows the top third of the films to undergo more 〈111〉 grain growth than 

the bottom two-thirds. The surface grains are most likely 〈111〉-oriented because the 〈111〉 

texture increases during annealing while the 〈100〉 texture remains the same. However, 

the larger grains do not necessarily have less strain than the other grains (other than that 

dictated by equation (3.30) if they are indeed 〈111〉 textured). The average film stress is ap- 

proximately 0 at 660 ◦ C. It is more likely that the strain relaxation is caused by dislocation 

emission through the free surface. 

The (111) peak of the thermally cycled 1.73 µm thick films was also viewed in the pow- 

der diffraction geometry. As described above, strong gradients may cause an asymmetry in 

this peak; however, no peak asymmetry in the (111) peak was discernible.


Anisotropy 

According the results based on the Schmid factor in Appendix D, the 〈111〉 textured 

grains should yield before the 〈100〉 grains. If we imagine that each grain undergoes the 

same plastic strain after the initial yield, the 〈111〉 grains will have undergone more plastic 

strain than 〈100〉 grains. Consequently, the elastic strain left will be in the order 

ɛ〈100〉 >ɛ〈111〉. (3.31) 

The results in Figure 3.9 agree with this analysis. This may not be correct, however. 

According to a study by Owusu-Boahen and King [179], at the very onset of yielding 

in thin films, dislocation nucleation at the surface takes place on apical slip planes (planes 

lying at the intersection of grain boundary triple junctions and the free surface) because 

a nearly infinitesimally short line segment can be nucleated here. Furthermore, the grain 

boundaries at the junctions must contain low energy grain boundary dislocations. Conse- 

quently, the Schmid factor is not a good predictor of which grains will slip first in this very 

early yield regime. 

A better explanation is that if the large grains near the surface are indeed 〈111〉 textured, 

it would be easier for these grains to undergo plastic relaxation simply because of the 

proximity to the surface. The dislocations in 〈100〉 textured grains may be constrained by 

the grain boundaries. 

The random texture component seems to have strains and strain profiles very similar to 

those of the 〈111〉 grains. Currently, it is not understood why this is.



According to the SGP theory, the strain profile in thermally cycled films should exhibit 

surface relaxation and a boundary layer at the substrate interface. Our GID measurements 

show that the surface relaxation indeed exists: proof of the boundary layer could not be ob- 

tained. The as-deposited films exhibit large tensile stresses at the surface while the average 

stress is compressive. 

Strain anisotropy in the as-deposited films is best explained by the anisotropy in the 

biaxial modulus. The strain anisotropy in the thermally cycled films may be due to a texture 

gradient along the surface normal. 

The best method for studying the boundary layer would be to perform an in situ diffrac- 

tion experiment on tensile-tested free-standing films with thin dislocation barrier layer (Ta 

or polyimide?) and very large plastic strains. A GID experiment may be performed through 

the thin barrier layer using very high x-ray energies and an energy sensitive detector. 

Strain gradients should also dominate the plastic behavior in multilayers [234]. These 

gradients may be studied using the methods discussed in the next chapter.

Chapter 4 

The Ag/Ni {111} Interface Stress 

4.1 The Ag/Ni {111} Interface 

Chapter 3 presented the Matthews-Blakeslee equation (3.9) for the maximum thickness 

of an epitaxial layer before the onset of dislocation nucleation. In much the same way, 

we can calculate a critical bilayer period for multilayers of equal phase thicknesses; the 

critical thickness of each layer is about four times greater than that for a film on an infinite 

substrate [65, 228]. This introduction provides background on the structure and energy 

of the {111}Ag/{111}Ni interface (hereafter called the {111} interface), as well as some 

requisite thermodynamics for deriving a relationship between interface energy and interface 

stress. Subsequently, I will describe the experiment performed to measure the interface 

stress and discuss its origin. 

82

Chapter 4: The Ag/Ni {111} Interface Stress 83 

4.1.1 The Structure and Energy of Semicoherent Interfaces 

Numerous reviews on the structure and energy of heterophase interfaces exist; an ex- 

cellent starting point is the book by Howe [104], and Sutton and Balluffi have recently 

completed a comprehensive treatise on solid interfaces [214]. I begin by first defining the 

following two parameters. The misfit between two materials α and β is given by 

and the spacing between misfit dislocations is 

δ = 2 � � 

aα − aβ 

, (4.1) 

aα + aβ 

Dδ = b 

, (4.2) 

δ 

where a is the lattice parameter, and b is the Burgers vector of the misfit dislocation. In 

general, knowledge of the misfit, the elastic constants of the adjacent materials, and the 

interface crystallography can provide an estimate of the interface structure and energy in 

an idealized system. For real systems, many other factors such as substrate surface defects 

and dislocation kinetics become important. 

Structure of the Ag/Ni {111} Interface 

Silver and nickel have negligible solid solubility at room temperature; the phase dia- 

gram is reproduced in Figure 4.1. They also have a large misfit of 14.8%, with aAg = 

0.40862 nm and aNi = 0.35238 nm [115]. For both these reasons, the structures of low- 

index Ag/Ni interfaces have been widely studied by both cross-sectional TEM [75, 154] 

and simulation [55, 74, 76, 87]. According to these studies, the {111} interface with 0 ◦ 

twist has the lowest interface energy. Figure 4.2 shows a view of this interface along 

the [11¯1] direction. The Ag atoms are black and the Ni atoms are white. According to


O-lattice theory [19], a {111} interface produces a hexagonal network (three sets of dislo- 

cations intersecting at 60 ◦ )of1/2〈110〉-type Burgers vectors. If we take our top view of 

the interface and cut along a (¯110), we obtain the cross-sectional view in Figure 4.3, which 

looks along the [110]. Two types of dislocation nodes may be seen: one is associated with 

{...CABC}Ni / {BC AB ...}Ag stacking and another with {...CABC}Ni / {CABC...}Ag 

stacking [76, 90]. The former is an intrinsic stacking fault (ISF), while the latter is a high 

energy fault. Since the energy of dislocations scales as b 2 (equation (1.9)), it is energet- 

ically favorable for these dislocations to dissociate into 1/6〈211〉 Shockley partials [76]. 

After dissociation, the misfit dislocations form the triangles seen in Figure 4.2; at the cen- 

ters of the white triangles lie planes with the correct stacking while the intrinsic stacking 

faults lie at the centers of the shaded triangles. The high energy faults lie at the nodes of 

the triangles. 

The cross-sectional TEM images showed that the misfit dislocation cores lie in the 

{111} interface and are slightly delocalized [75]. Other Ag/Ni interfaces, such as the 

{110}Ag/{110}Ni and the {100}Ag/{100}Ni are less closely packed than the {111} interfaces 

and allow misfit dislocations to move out of the interface and into the Ni one plane deep 

[260]. For the {110}Ag/{110}Ni interface, Gumbsch et al. [89] have found that this expulsion 

of misfit dislocations can lead to a collection of vacancies on the Ni side of the boundary, 

which lowers the interfacial enthalpy. This could potentially happen at the high energy 

fault positions in the {111} interface, but the energy decrease in the removal of the fault is 

less than the cost of vacancy formation [89].


Figure 4.1: The Ag/Ni phase diagram [149].


Figure 4.2: Top view of the Ag/Ni {111} interface, with Ag and Ni denoted by closed and 

open circles, respectively. After relaxation, misfit dislocations are formed, as delineated by 

the dark lines. The coherent areas and intrinsic stacking fault areas lie in the centers of the 

white and shaded triangles, respectively.


Figure 4.3: An edge-on view of the Ag/Ni {111} interface along the [110] direction, with 

the Ag and Ni atoms denoted by closed and open circles, respectively. The large circles are 

atoms in the plane of the diagram; the small circles are atoms in the planes immediately 

above and below the plane of the figure. Two distinct dislocation nodes can be seen and 

associated with the different stacking conditions. Adapted from Gao and Merkle [75].


Energy of the Ag/Ni {111} Interface 

Turnbull suggested that the energy of a semicoherent interface may be calculated as the 

sum of chemical and structural effects for small misfit [226]: 

γ = γc + γs 

(4.3) 

where γ is the energy of the solid-solid interface, and the subscripts c and s refer to the 

chemical and structural components. The chemical component can be estimated with var- 

ious theories (e.g., Cahn-Hilliard [27, 28] or the discrete lattice plane model [139]), while 

the structural component has been described using the dislocation models of Frank-van der 

Merwe [63, 64] and Matthews [150]. A recent paper by Willis et al. [250] demonstrates the 

equivalence of these two dislocation models when the spacing between the misfit disloca- 

tions goes to infinity. 

For simplicity, only the Matthews result is presented. The energy of two perpendicular 

and noninteracting arrays of edge dislocations is 

� � � � 

(δ − εcoh) R 

GfGs 

γ = b ln + 1 

π b (Gf + Gs)(1 − ν) 

� �� 

ξ 

Mcomp 

(4.4) 

where G is the shear modulus, ν is Poisson’s ratio of both the film and substrate (assumed 

equal), R is the cut-off parameter, and εcoh is the coherency strain between 0 and δ. The 

variable ξ is a dimensionless parameter representing the behavior of the stress field near the 

dislocation core, and Mcomp is the composite modulus. Refinements on this equation that 

consider anharmonic effects [155, 147, 146] as well as more complex dislocation arrays 

are available [20, 21].


4.2 Interfacial Thermodynamics 

This section is based mainly on Sutton and Balluffi [214] who, in turn, base most of 

their material on Cahn [26]. An interesting and rigorous treatment of the thermodynamics 

of solid surfaces is provided by Rusanov [193]. 

4.2.1 Fluid-Fluid Interfaces 

For planar fluid-fluid interfaces, a change in internal energy can be expressed as 

dU = TdS− PdV + µidNi + γ dA (4.5) 

where U is the internal energy, T is the temperature, S is the entropy, P is the hydrostatic 

pressure, V is the volume, µi is the chemical potential per atom, Ni is the number of 

atoms of species i, γ is the interface free energy, A is the interface area, and the Einstein 

summation notation is used. Since the internal energy is a homogeneous function of the 

first order, 

or 

Taking the total differential of equation (4.6) gives 

U = TS− PV + µi Ni + γ A (4.6) 

γ = 1 

A (U − TS+ PV − µi Ni) . (4.7) 

dU =TdS− PdV + µidNi + γ dA+ (4.8) 

SdT − VdP + dµi Ni + Adγ.


After subtracting equation (4.5) from (4.8), we obtain 

or 

0 = SdT − VdP + Nidµ + Adγ (4.9) 

dγ =−[S]dT + [V ]dP − [Ni]dµ. (4.10) 

The bracketed variables are the so-called layer quantities [26] and are not uniquely defined. 

The Gibbs-Duhem equations for the two fluid phases α and β are 

0 = −S α dT + V α dP − N α 

i dµi 

0 = −S β dT + V β dP − N β 

i dµi. 

(4.11) 

Using Cramer’s rule to solve (4.10) and (4.11) for dγ , we obtain the Gibbs adsorption 

equation for fluid interfaces 

dγ =−�SdT + �VdP − �Nidµi 

(4.12) 

in which the excess quantities �Xi refer to the extra X per unit area compared to a system 

with the same composition α and β phases but no interface. The excess quantities are 

well-defined for any choice of the dividing surface. 

4.2.2 Solid-Fluid Interfaces 

Since surface atoms have different bonding and average coordination from those in the 

bulk, the surface layer has a different lattice parameter than the bulk. However, the surface 

is coherent with the bulk (ignoring surface dislocations and reconstructions), so a stress 

must be present in the layer.


Here, I closely follow Sutton and Balluffi’s derivation of the relationship between sur- 

face stress and surface free energy [214]; Shuttleworth’s original derivation can be found 

in his paper [206] or in Cammarata’s review [29]. 

We can imagine a free-standing slab of thickness t, length L, and width L, where L ≫ t 

(Figure 4.4). In this schematic, the two surfaces are stretched to fit onto the bulk, much in 

the same way as the film was stretched to fit onto the substrate in Figure 1.3. If the slab is 

thick enough, we can approximate the stress distribution in the bulk as uniform through its 

thickness resulting in the force balance relationship 

fij = −σijt 

2 

for i, j = 1, 2 (4.13) 

where fij is the force on the surface per unit width (surface stress), σij is the bulk stress, 

and I have already inserted the generalized tensor form for f and σ , which can easily be 

proven [159]. 

Now a variational strain δɛij is applied to the slab. The change in free energy is given 

by the change in surface free energy plus the change in bulk strain energy: 

� 

∂(2γ L2 ) 

δG = 

∂ɛij 

� 

2 ∂w 

+ tL δɛij, (4.14) 

∂ɛij 

where w is the strain energy density of the bulk and related to the stresses by 

∂w 

∂ɛij 

= σij. (4.15) 

Under equilibrium conditions, δG = 0. After inserting in equation (4.13) and noting that 

the δɛij are independent variations, we obtain 

fijA = γ 

∂ A 

∂ɛij 

+ A ∂γ 

. (4.16) 

∂ɛij


Finally, since 

∂ A 

∂ɛij 

we arrive at the Shuttleworth relationship: 

= A for i = j (4.17) 

= 0 for i �= j, (4.18) 

fij = δijγ + ∂γ 

∂ɛij 

. (4.19) 

We can now add the surface stress term to the Gibbs adsorption equation: 

dγ =−�SdT + �VdP − �Nidµi + ( fij − δijγ)dɛij. (4.20) 

An interesting Maxwell relation between the surface stress and the excess volume can be 

obtained by taking second derivatives of γ and changing the order of differentiation: 

� � 

∂( fij − δijγ) 

∂ P 

� � 

∂�V 

= 

. (4.21) 

4.2.3 Solid-Solid Interfaces 

T,µk 

Solid-solid interfaces can be treated in a similar way, although one needs to be care- 

ful about distinguishing between coherent and incoherent interfaces. Figure 4.5 shows a 

∂ɛij 

T,µk 

similar schematic to Figure 4.4, but now applied to solid multilayers. 

In analogy to the previous section, the Shuttleworth relationship for solid-solid inter- 

faces is 

fij = f α β 

ij + fij = δijγ + ∂γ 

. (4.22) 

∂ɛij 

This equation is for a coherent interface. For an incoherent interface, one phase may be 

strained relative to the other one by an amount ɛ r ; we then require two interface stresses


t 

Figure 4.4: A schematic demonstrating how misfitting surface layers can lead to internal 

bulk stresses. 

t 

t 

β 

α 

β 

α 

Figure 4.5: A schematic showing how interface stresses can lead to internal bulk stresses. 

α 

f11 α 

f11 α 

f11 α 

f11 β 

σ11 f 11 

f 11 

α 

σ11 β 

σ11 α 

σ11 σ 11 

β 

f11 β 

f11 β 

f11 β 

f11


[30], 


fij = δijγ + ∂γ 

∂ɛij 

gij = δijγ + ∂γ 

∂ɛ r ij 

(4.23) 

(4.24) 

where we can now connote fij with straining a unit area at constant interface structure 

and gij with changing the interface structure with different strains in the two phases but 

constant average strain [247]. 

The Maxwell relationship between the excess volume and interface stress is the same 

as that in equation (4.21) if the interfaces are incoherent; for coherent interfaces, no simple 

relationship has yet been derived, although Johnson et al. [117, 118, 116] have made sig- 

nificant progress. The problem is that the Gibbs-Duhem equation needs to be reformulated 

when the system contains coherency stresses [137, 117]. Johnson found that for coher- 

ent interfaces, the interface stress and excess volume were physically meaningful only for 

certain multilayer compositions [116]. This result has not yet been corroborated. 

The general theory relating the interface stress to the average volume stress was pre- 

sented by Weissmüller and Cahn [247]. Their result as applied to single crystal multilayers 

is presented here: 

〈σij〉v = −2 

� 

⎡ ⎤ 

⎢ f11 

⎢ f12 ⎢ 

⎣ 

f12 

f22 

0 ⎥ 

0⎥ 

, 

⎥ 

⎦ 

(4.25) 

0 0 0 

where 〈σij〉v is the average volume stress tensor and � is the bilayer thickness. For a biaxial


stress state and interfaces with three-fold symmetry or higher, this reduces to 

〈σ 〉v = −2 

� 

f (4.26) 

where f is now a scalar. Weissmüller and Cahn note that this equation does not assume 

independently relaxed strains in the individual layers. Also, the interface stress gij does 

not contribute to to the average volume stress. 

Prior to Weissmüller and Cahn, Ruud et al. [196] derived a similar equation for use in 

measuring the Ag/Ni interface stress. The Ag and Ni layers were not single crystals so their 

value of f is a combination of the Ag/Ni interface stress and Ag and Ni grain boundary 

stresses. Spaepen gives a correction in Ref. [212]. An independent determination of the Ag 

and Ni grain boundary stresses has not yet been carried out. 

4.3 Principle of the Interface Stress Measurement 

In Chapter 1, we discussed several “types” of stresses in thin films and categorized 

them according to their measurability by wafer curvature and x-ray methods (Table 1.1). 

If the thicknesses of the Ag and Ni layers are equal, the substrate curvature change due to 

coherency stresses is zero [101]; in this case, barring any interface phase formation, the 

difference between the curvature measured stress and the x-ray measured stress originates 

from the interface stress. 

For a multilayer on a substrate, equation (4.26) becomes 

� � 2 Nf 

σsc −〈σ 〉x−ray = f = 

� tf 

(4.27) 

where σsc is the stress on the multilayer due to the substrate constraint, 〈σ 〉x−ray is the 

average volume stress measured by x-rays, tf is the total film thickness, and N is the total


Table 4.1: The Ag/Ni {111} Interface Energy 

Chemical Structural Total Preparation Method Ref. 

- - 0.73 ± 0.12 N/m e-beam (725 ◦ C) [121] 

0.05 N/m ∗ 0.37 N/m 0.42 N/m EAM calculation [87] 

∗ Average of the coherent stacking and intrinsic stacking fault areas. 

number of interfaces [196]. By plotting � � 

σsc −〈σ 〉x−ray versus �−1 and performing a 

linear fit, the interface stress can be obtained from the slope. 

4.4 Previous Measurements of the Ag/Ni {111} Interface 

Energy and Stress 

There has been only one measurement of the Ag/Ni {111} interface energy (Table 4.1) 

and four of the Ag/Ni {111} interface stress (Table 4.2), including the work presented here. 

The measured interface energy is larger than the embedded atom method (EAM) result, 

but this may be partially attributable to the temperature dependence of the interface energy 

[211] since the EAM calculation was performed at zero temperature. According to the 

EAM result, the structural component dominates the interface energy. 

Three of the four Ag/Ni interface stress measurements rely on equation (4.27) and one 

relies on equation (4.26) [120]. The multilayers in each of the four experiments vary in 

deposition parameters as well as layer quality; however, these differences have little effect 

on the value of the interface stress. The other FCC metallic multilayers listed in Table 4.2 

have similar misfits to Ag/Ni, and their interface stresses are similar in sign and magnitude 

to that of Ag/Ni. All of these experimental results differ significantly from the EAM results,


∗ Dynamic value 

Table 4.2: {111} Interface Stress Results for FCC Metallic Multilayers 

System Value Preparation Method Ref. 

Ag/Ni −2.02 ± 0.26 N/m e-beam (∼ −196 ◦ C) [120] 

Ag/Ni −2.24 ± 0.21 N/m sputtering (−2 ◦ C) [200] 

Ag/Ni −2.27 ± 0.67 N/m sputtering (−156 ◦ C) [196] 

Ag/Ni −2.17 ± 0.15 N/m sputtering (−190 ◦ C) this work 

Ag/Ni +0.32 N/m EAM calculation [88] 

Ag/Cu −3.19 ± 0.43 N/m sputtering (35 ◦ C) [14] 

Ag/Cu −0.21 ± 0.10 N/m ∗ evaporation (40 ◦ C) [205] 

Ag/Cu +0.32 N/m EAM calculation [88] 

Au/Ni −2.69 ± 0.43 N/m † sputtered (−5 ◦ C) [198] 

Au/Ni −0.08 N/m EAM calculation [88] 

† After accounting for intermixing 

which are smaller in magnitude and are generally tensile. 

To resolve the discrepancy between the EAM results and experiment, we consider the 

following two scenarios. The first scenario is that the interpretation of the experiments 

is flawed. Clemens et al. [39] have hypothesized that intermixing between the multilayer 

components interferes with the interface stress measurement. The second scenario is that 

the interpretation is correct, and the discrepancy with the EAM calculations arises from the 

inability of EAM to consider stresses in a real system. For instance, the EAM calculation 

was performed for bilayers with very small residual strains at a temperature of zero Kelvin, 

while the experiments measure f in multilayers with large strains at room temperature. 

Furthermore, the embedded atom method may only apply to the linearly elastic regime 

(only the second order elastic constants were used as inputs) [45]. 

Two experiments were carried out to help shed light on the controversy. One remeasures 

the Ag/Ni {111} interface stress. The other examines the strain and compositional profiles 

of the multilayers. After presenting the results, I will return to the discrepancy in section


4.7. 



Ag/Ni multilayered films were deposited by ion beam sputtering 99.99 at.% Ag and 

99.99 at.% Ni targets using a gettered Ar sputtering gas. The chamber used for deposition 

is shown in Figure 1.7; it includes a rotating target block to switch between the Ag and 

Ni targets [213], which eliminates the need for precise shuttering. The Si(100) substrates 

(381 µm thick) were kept near −190 ◦ C during deposition to prevent balling up of Ag on Ni 

and vice versa, as well as to minimize interdiffusion. The temperature was monitored with 

a thermocouple placed near the backside of the substrate (the side away from the deposition 

flux). Five different bilayer thicknesses were made: 3.2 nm, 5.9 nm, 11.3 nm, 17.3 nm, and 

22.8 nm. All sputtering targets and substrates were sputter cleaned prior to deposition. The 

sputtering parameters are given in Table 1.2. 

Subsequent to the interface stress measurement, another multilayer with � = 4.2nm 

was grown for specular reflection and diffuse scattering measurements. The results for this 

multilayer are given in section 4.7.1. 


Rutherford Backscattering Spectrometry (RBS) was used to determine the film compo- 

sition and the overall thickness of the multilayers. The thicknesses were also checked with 

a Tencor Alpha-Step 200 Profilometer. The film quality was determined by both low-angle


x-ray reflectivity and cross-sectional transmission electron microscopy (TEM). 


The stress applied by the substrate on the multilayer was determined by measuring the 

change in curvature with the wafer curvature apparatus (Figure 1.4) described in Chapter 1; 

the Stoney equation (1.16) was used to convert the curvatures into stresses. Measurements 

on a laser scanning apparatus [257] were taken both before and after deposition. Since 

substantial stress relaxation took place after deposition, the x-ray scans were performed 

only after the stress stabilized. 


The multilayers were examined at the National Synchrotron Light Source at beamline 

X22C with a wavelength of 0.154591 nm; some measurements were also performed with a 

shorter wavelength, 0.149020 nm. The wavelengths were chosen to be far from and close 

to the Ni K absorption edge to effect a large change in f ′ , the real part of the atomic 

scattering factor. Low-angle θ − 2θ scans were made to quantify the bilayer thickness and 

interface roughness, and high-angle scans were taken to measure the out-of-plane strains. 

The SUPREX software package [70] and Bede Scientific’s REFS software were used to 

analyze the high-angle and low-angle θ − 2θ results. 

The in-plane strains, ɛ�, were measured with grazing incidence diffraction (GID) [148] 

at incidence angles of 1 ◦ and 2 ◦ . The {220} diffraction planes for silver and nickel were 

chosen for the measurement using the bulk lattice parameters of Ag and Ni for reference.


The average volume stress was then calculated using 

〈σ 〉x−ray = YAg tAg ɛ�,Ag + YNi tNi ɛ�,Ni 

tAg + tNi 

(4.28) 

where Yi is the biaxial modulus, ti is the layer thickness, and ɛ�,i is the in-plane strain of 

element i. Due to the large strains involved, Yi is strain dependent, and higher order elastic 

constants are required. Assuming perfect {111} texture, we use the formulas [196, 240] 

4.6 Results 


YAg(ɛ) = 173.60 − 3764.9 ɛ − 16349 ɛ 2 (GPa) (4.29) 

and YNi(ɛ) = 389.32 − 4903.9 ɛ − 17006 ɛ 2 (GPa). (4.30) 

Figure 4.6 shows a cross-sectional TEM micrograph of a � = 4.2 nm Ag/Ni multilayer. 

The layers are of good quality and have well-defined interfaces, as confirmed by the low- 

angle diffraction scans shown in Figure 4.7. Fitting of the low-angle data gave an average 

total interface roughness of 0.4 nm for all of the multilayers used in the interface stress 

measurement. 

The RBS data indicated that the films were slightly Ag rich (51-53 at.%), and the total 

thicknesses varied from 1.14 to 1.29 µm. From these results, the deposition rates were 

calculated to be approximately 0.5nm/s for Ag and 0.1nm/s for Ni.


Figure 4.6: A cross-sectional view of a Ag/Ni multilayer with � = 4.2nm. 

4.6.2 Substrate Curvature Stress 

The stress of the multilayer due to the substrate constraint is shown as the open circles 

in Figure 4.8 (b). In general, the magnitude of σsc increases slightly with bilayer thickness. 

4.6.3 Volume Stress 

FCC metallic multilayers are expected to have 〈111〉 texture; the high-angle θ − 2θ 

scans confirmed this and are shown later. Since most of the grains are 〈111〉-oriented, 

the macroscopic in-plane biaxial strain can be determined by the d-spacings of the {220} 

diffraction peaks. 

The Ag {220}, Ni{220}, Ag{311}, and Ag {222} GID peaks in Figure 4.9 were simul- 

taneously fit with four pseudo-Voigt functions [248]. The resulting strains are shown in


Λ = 11.3 nm 

Λ = 5.9 nm 

Λ = 3.2 nm 

Figure 4.7: Offset low-angle θ − 2θ diffraction spectra for the three smallest bilayer thicknesses. 

The spectra were taken with an x-ray energy of 8020 eV.


σ 

Λ 

Λ 

〈σ〉 

σ 

(a) (b) 

Figure 4.8: (a) Volume stresses in Ni (closed squares) and Ag (open squares) layers as 

a function of inverse bilayer thickness. (b) Substrate curvature stress (open circles) and 

average volume stress (closed circles) as a function of inverse bilayer thickness. The error 

bars for all the data are within the size of the symbols. The arrow drawn from (a) to (b) 

illustrates how the layer stresses are averaged to give 〈σ 〉x−ray. 

Λ 

Λ


Λ 

θ [degrees] 

Figure 4.9: Offset GID spectra for Ag/Ni multilayers having the indicated bilayer thickness 

�. The vertical lines indicate the {220} peak positions for strain-free bulk Ag and Ni. The 

x-ray energy was 8020 eV. 

Figure 4.10. 

The strains were converted to volume stresses with the use of higher order elastic con- 

stants, as discussed earlier. Figure 4.9 shows that the Ni {220} and Ag {311} peaks overlap 

significantly for the � = 17.3 nm and 22.8 nm samples, preventing an accurate measure of 

〈σ 〉x−ray at these bilayer thicknesses. These samples were not used when calculating the 

interface stress. 

The average volume stresses calculated using equation (4.28) are shown as the closed 

circles in Figure 4.8 (b). This figure also shows that σsc and 〈σ 〉x−ray converge as the 

bilayer thickness increases. Without an interface stress, these curves should coincide for 

all bilayer thicknesses.


ε 

Λ 

Λ 

Figure 4.10: In-plane strains in nickel and silver as a function of bilayer thickness. 

4.6.4 The Interface Stress 

The difference between the substrate curvature stress and the average volume stress is 

plotted as a function of inverse bilayer thickness in Figure 4.11. A weighted least squares 

fit for the three smallest bilayer thicknesses produces a line with slope −4.34, giving an 

interface stress of −2.17 ± 0.15Nm −1 . The interface stress appears to increase sharply 

when �>11.3 nm for unknown reasons but is most likely due to a change in the degree 

of coherency of the interface. Schweitz et al. [200] noted a similar trend in their data. 

This value agrees with the previous results listed in Table 4.2. As mentioned, the inter- 

face stress is large and compressive while the EAM result is small and tensile. 

Before going on to discuss different models for the compressive interface stress, I repeat


σ 〈σ〉 

Λ 

Λ 

Figure 4.11: Difference between the substrate curvature stress and the average volume 

stress for Ag/Ni multilayers as a function of inverse bilayer thickness. The line is a 

weighted least-squares fit for the three smallest bilayer thicknesses.


the Shuttleworth relation, 

f = γ + ∂γ 

. (4.31) 

∂ɛ 

The first term on the right side is the excess energy per unit deformed area, i.e., the area 

after the interface has been deformed to fit between the bulk phases [174]. It is possible for 

γ to be negative, although not for a stable interface; however, experiment and calculations 

agree that it is positive. Then for f to be negative, the derivative term must be negative, 

with the interface energy decreasing with increasing area. This result is not intuitive and has 

led some to question the validity of the experiment. Note that the EAM simulation actually 

shows the same phenomenon, although the magnitude is much smaller; γ = 0.42 N/m 

with f = 0.32 N/m. 

4.7 Models for the Compressive Interface Stress 

Here, I return to the questions concerning the disagreement between the EAM results 

and experiment and investigate the possibility that the interpretation of the experiments is 

flawed by intermixing. Afterwards, I will discuss another possibility: the interpretation of 

the experiments is correct and the EAM calculations do not consider an analogous system 

that merits comparison. 

4.7.1 Intermixing 

Nickel appears to have a negative Poisson ratio in the plane of the Ag/Ni multilayers 

[190, 200]. Although the in-plane strains are strongly tensile, the out-of-plane strains are 

also tensile, as shown in Figure 4.12. This is very different from the single crystal value.


Λ 

Figure 4.12: The average {111} d-spacing for nickel. Since the nickel is in tension parallel 

to the {111} planes, d111 is expected to be less than the strain-free d-spacing of 0.20345 nm 

for a positive Poisson ratio. 

Λ 

Gergaud et al. [83], using the sin 2 ψ-method (Appendix A), have claimed that the 

strain-free lattice parameter of Ni increases with smaller bilayer thicknesses, and attributes 

the increase to intermixing. Use of the sin 2 ψ-method, however, should be used with cau- 

tion when working with multilayer structures. Chocyk et al. [38] have demonstrated that 

the sin 2 ψ-method gives incorrect stresses when applied to the Au/Ni multilayers. Further- 

more, as shown in Appendix A, this method of determining the strain-free lattice parameter 

relies on the bulk Poisson ratio; hence the negative Poisson ratio effect and the strain-free 

lattice parameter problem are related. The question is whether both can be explained by a 

self-consistent model. 

In view of Gergaud’s results, Clemens et al. [39] reasoned that since the surface energy 

of Ag is lower than that of Ni, Ag atoms prefer to lie on the film surface. If, during


Ni deposition, the deposition kinetics were such that a substantial amount of Ag could 

segregate into the Ni and become kinetically trapped, this would invalidate the use of the 

nickel bulk lattice parameter when calculating the strain. Consequently, this would yield a 

spurious value for the interface stress. 

Intermixing, however, should be dependent on the deposition kinetics (Figure 4.13). If 

the segregation of Ag through Ni is too slow, this results in no mixing. If the segregation 

is too fast, then Ag “floats” through the Ni and does not mix. If the segregation falls into 

an intermediate regime, most likely a compositional gradient exists between the Ni and Ag 

layers; it seems unlikely that Ag can homogenize itself in the Ni during Ni deposition to 

achieve uniform mixing. Recall that our experiments were performed at liquid nitrogen 

temperatures. If the gradient is too steep, there only exists incoherent scattering from the 

alloyed planes. The intensity due to incoherent scattering is too weak to contribute to the 

Ni {220} diffraction peak and affect the interface stress [196]. If the gradient is too shallow, 

this would result in an alloy peak between the Ag and Ni {220} peaks in Figure 4.9. This is 

not seen. 

In the case of uniform or nearly uniform intermixing, Clemens et al. [39] found that 

equation (4.27) should be replaced by 

where 

δmix = 1 

� 

YAg 

2 

� � 2 

σsc −〈σ 〉x−ray = 

� ( f + δmix) (4.32) 

(dAg, 0 − dNi, 0) 

(dAg, 0 − dNi, 0) 

tNi into Ag − YNi 

tAg into Ni 

dAg, 0 

dNi, 0 

� 

(4.33) 

assuming a linear relationship between the degree of alloying and the change in lattice 

parameter (Vegard’s law). Here, di, 0 is the strain-free lattice parameter of element i, and


Ni 

Ag 

cold hot 

Figure 4.13: Depiction of the kinetics of segregation. At cold temperatures, the Ag lacks the 

mobility to segregate. At high temperatures, the Ag “floats” to the surface. At intermediate 

temperatures, a compositional gradient exists. 

tα into β is the the equivalent thickness of α mixed with β. According to their argument, 

δmix is much larger than f and is consistently measured to be −2N/m. 

The composition gradient can be estimated using the following method of determining 

the strain and composition profiles. 

A Systematic Method of Extracting Strain and Composition Profiles of Multilayers 

The strain and composition profile of a multilayer can be obtained from RBS and θ −2θ 

scans at both low and high angles. The low-angle data result from optical interference of 

the reflected rays from the multilayer and can be simulated using an optical multilayer 

formalism [85]. Hence, low-angle reflectivity is often performed on both amorphous and 

crystalline multilayers. The high-angle data result from Bragg diffraction off a superlattice 

[70]. The data from each are complementary. For example, both give a bilayer thickness, 

but the high-angle data also provide d-spacings while the low-angle data provide mass


N α 

N β 

Λ 

d α 

d β 

(111) 

(111) 

Figure 4.14: Depiction of a superlattice unit cell using the parameters in equation (4.34). 

densities. Agreement between the two on the bilayer thickness allows a consistency check. 

However, it should be remembered that the high-angle data provide a bilayer thickness 

based on Bragg diffraction; for a single crystal multilayer, the agreement should be perfect, 

butifthefilm is polycrystalline, the bilayer thicknesses may not match exactly, especially 

if a texture gradient exists in the multilayer. 

The first step of the method is to use RBS to find the ratio Nα/Nβ, where Nα and Nβ 

are the number of α and β monolayers in one repeat unit, �. Now we turn to the high-angle 

θ − 2θ data. The superlattice unit cell is given by 

� = Nαdα 

� �� 

tα 

+ Nβdβ 

� �� 

tβ 

where the parameters are defined in Figure 4.14. Bragg’s law for a superlattice is 

2 sin θ 

λ 

= 1 

¯d 

± n 

� 

t α 

t β 

(4.34) 

(4.35)


Figure 4.15: High-angle x-ray diffraction peaks for a Ag/Ni multilayer with � = 4.2 nm. 

The order of the satellite is given above its peak. The data taken at E = 8320 eV are offset 

from the E = 8020 eV data. 

where n is the order of the satellite and 

¯d = 

� 

Nα + Nβ 

. (4.36) 

By plotting 2 sin θ/λ vs. n, we can determine ¯d and � by fitting a straight line. Since ¯d is 

given by (4.36) and we know Nα/Nβ from RBS, we can solve for Nα and Nβ. 

At this stage, we know �, Nα, and Nβ but have no information on dα and dβ. Arefine- 

ment method can be used to determine these parameters. Figure 4.15 shows an example 

of a fit using SUPREX [70]. In addition to providing dα and dβ and therefore tα and tβ, 

SUPREX supplies additional information I will discuss later. 

We now use the low-angle θ −2θ data. Details on the theory behind specular reflectivity


and diffuse scattering can be found in a recent text by Als-Nielsen and McMorrow [2]. 1 

With tα and tβ used as consistency checks, true specular reflectivity profiles 2 are fit in order 

to determine the total interface roughnesses, σtotal, of both the Ag/Ni and Ni/Ag interfaces. 

As shown in Figure 4.16, specular reflectivity samples only along qz in reciprocal space and 

cannot distinguish between physical and chemical roughness. Longitudinal and transverse 

diffuse scattering scans are then recorded and simulated. The transverse scans sample 

qx information at fixed qz, where the x direction is parallel to the sample surface, and 

longitudinal scans sample reciprocal space in both x and z directions just off the specular 

“ridge” at qx = 0. By having the qx information, a distinction between the physical and 

chemical roughness can be made; they are related to the total interface roughness by 

σ 2 

total = σ 2 physical + σ 2 chemical . (4.37) 

The diffuse scattering profile simulations use the Distorted-Wave Born Approximation 

(DWBA) [209, 138]. An example of a specular reflectivity fit and a longitudinal diffuse 

scattering simulation are presented in Figure 4.17 (a) and (b), respectively. 

Finally, the multilayer can be studied with different x-ray energies, as was done in this 

experiment. Two independent sets of data can be taken if the energies induce scattering 

with different atomic scattering factors. This provides another consistency check for all the 

results. 

Results from Diffuse Scattering 

The systematic method described above was performed on the � = 4.2 nm sample. 

Only the diffuse scattering result is presented, as the other results are similar to those of the 

1 Goldman et al. [85] demonstrate how the specular data can be analyzed systematically. 

2 The remaining profile after the subtraction of a longitudinal diffuse scan from a specular reflectivity scan.


q z 

0 

Specular scan 

Transverse scan 

0 

∆θ 

q x 

Longitudinal scan 

Figure 4.16: An illustration of a reciprocal space map along qx and qz. The specular ridge 

samples along qz, transverse scans sample along qx, and longitudinal scans sample both 

components.


∆θ = 0° 

∆θ = 0.55° 

Figure 4.17: Specular reflectivity profiles (a) and longitudinal diffuse scattering profiles 

(b). The tilt of the sample from the specular condition is indicated by �θ, and the x-ray 

spectra taken at E = 8020 eV and E = 8320 eV are offset. The data and fitted/simulated 

profiles are given by the open circles and solid (or dashed) curves, respectively. The solid 

curves show the results of a simulation with a Ag/Ni (surface side / substrate side) chemical 

roughness of 0.24 nm and a Ni/Ag chemical roughness of 0.30 nm. The offset dashed 

curves show the results of a simulation with a Ag/Ni chemical roughness of 0.30 nm and a 

Ni/Ag chemical roughness of 0.40 nm.


other multilayers and are discussed in the next section. 

By simulating the diffuse scattering scans (Figure 4.17 (b)), it became evident that some 

chemical grading must be present at both interfaces; very high order harmonics would exist 

otherwise, even with large physical roughness parameters. However, an upper bound can be 

determined since too much chemical roughness can eliminate all the harmonics. The offset 

dashed curves in Figure 4.17 (b) show the effect of extra chemical roughness added to both 

the Ag/Ni and Ni/Ag interfaces. Specifically, the chemical roughness of Ag/Ni (surface 

side / substrate side) was changed from 0.24 nm to 0.30 nm, and the chemical roughness 

of Ni/Ag was changed from 0.30 nm to 0.40 nm. The additional chemical roughness in the 

latter simulation eliminates the third harmonic in the 8020 eV scan. By performing several 

such simulations, we determined an upper bound of σchemical ∼ 0.3 nm and a lower bound 

of σchemical ∼ 0.2 nm for both the Ag/Ni and Ni/Ag interfaces. 

Ruud et al. [196] considered a situation in which two interfacial {111} planes consist 

of Ni alloyed with Ag. With the use of Vegard’s law, the composition of these two planes 

producing the largest peak shift was calculated. This shift, however, corresponded to a 

change in the lattice parameter that was considerably smaller than the standard deviation of 

the lattice parameter measurement. Our upper and lower bounds of σchemical indicate that 

one to two intermixed {111} planes is reasonable; therefore, it is unlikely that intermixing 

affects our measurement of the interface stress. 

According to Josell et al. [120], even when the strain-free lattice parameters extracted 

from the sin 2 ψ-method are used, the interface stress remains compressive, with a value of 

∼−0.75 N/m. 

The remainder of this chapter is devoted to other rationales for a compressive interface


stress. 

4.7.2 Excess Volume / Misfit Dislocations 

We now return to some of the additional information supplied by SUPREX. The fits 

allow an extraction of the interface width (Figure 4.18). We see that the width of the 

Ni/Ag interface is close to the average {111} d-spacing of Ni and Ag. This is expected 

for a coherent interface. The Ag/Ni interface width is larger, however, by approximately 

0.02 nm. This distance is related to the excess volume per unit area discussed earlier. 

The SUPREX simulations also show that strain gradients are needed to fit the data. 

SUPREX allows the three planes adjacent to each side of the interface to deviate from 

the “bulk” d-spacing as shown schematically in Figure 4.19. The SUPREX fits for the 

� = 3.2 nm, � = 4.2 nm, and � = 5.9 nm multilayers all show the presence of steep 

d-spacing gradients within both the Ag and Ni layers. The multilayers with larger � have 

a significant overlap between the high-angle satellites and could not be fit. Figure 4.20 

shows the significance of the strain gradients. A model disallowing the gradients (the 

dotted curves) cannot reproduce the +3 satellite in the � = 3.2 nm sample or the +3 

and +4 satellite in the � = 5.9 nm sample while the model including the gradients fits the 

data well. Since our diffuse scattering results showed minimal interdiffusion, we assume 

the d-spacing gradients to be strain gradients rather than compositional gradients. This 

allows the construction of the d111 profiles in Figure 4.21. 

The d111 profiles resemble the results of Jaszczak et al., in which they simulated the 

strain profiles of (100)-oriented FCC multilayers with as misfits of 10% [113] and 20% 

[114] using the Lennard-Jones potential (Figure 4.21 (c)). If we interpret the Ag/Ni inter-


growth 

Ag (111) 

d 111, interface 

Ni (111) 

Λ 

Λ 

(a) 

(b) 

Figure 4.18: (a) A schematic showing the width between Ag and Ni (111) planes at 

the interface. (b) The interface width for Ag/Ni and Ni/Ag interfaces. The mean (111) 

d-spacing, 0.2197 nm, is indicated by the horizontal line. (d111,Ag = 0.23592 nm, and 

d111,Ni = 0.20345 nm).


Nα 

∆d α2 

d α + ∆d α2e 

d α + ∆d α2e 

d α 

d α + ∆d α1e 

d α + ∆d α1e 

∆d α1 

Figure 4.19: Schematic showing the method by which SUPREX treats gradients in the 

out-of-plane d-spacing. 

Figure 4.20: High-angle x-ray diffraction peaks for the � = 3.2nm(a) and � = 5.9nm 

(b) Ag/Ni multilayers. The intensity axis is on a logarithmic scale, and the data taken 

at E = 8320 eV is offset above the E = 8020 eV data. The circles are the measured 

intensities, the solid curve is the fit for the strain gradient model, and the dotted curve is the 

best fit when strain gradients are disallowed. The order of the reflection is indicated above 

the peak. 

−ξ 

−2ξ 

−2ξ 

−ξ





Si 

0.18 

0.20 

0.22 

d 111 [nm] 

0.24 

0.26 




0.18 

0.20 

0.22 

d 111 [nm] 

(a) (b) 

Si 

0.24 

0.26 

coherent 

0.14 

incoherent 

0.16 

0.18 

d 100 [nm] 

Figure 4.21: Calculated {111} d-spacing profiles for the � = 3.2nm (a) and 5.9nm (b) 

Ag/Ni multilayers. The results of Jaszczak et al. for their theoretical multilayer are provided 

in (c) [113]. The vertical dashed lines are the strain-free d-spacings for the bulk phases. 

(c ) 

0.20 

0.22


faces using Jaszczak’s results, this suggests the Ni/Ag interface is coherent while the Ag/Ni 

is semicoherent. 

According to the original Matthews-Blakeslee criterion, equation (3.9), there should 

exist no coherency between Ag and Ni; it predicts a coherent monolayer up to 9% mismatch 

[18]. However, Markov and Milchev [155, 147] have shown that adding anharmonic terms 

to the potential allows coherency up to 13.5% mismatch in a one-dimensional system. This 

emphasizes the importance of the anharmonic terms. 

The asymmetry of the interfaces most likely derives from the sign of the misfit [155, 

147]. Ni/Ag has a negative misfit with Ni having to stretch to fit onto Ag, while Ag/Ni has 

a positive misfit with Ag having to compress to fit onto the Ni. It should be easier to stretch 

than to compress because of the asymmetry of the potential well (Figure 1.1). Indeed, 

a room temperature growth experiment of Ni on Ag(111) probed by low energy electron 

diffraction (LEED) and Auger electron spectroscopy (AES) showed epitaxial growth up 

to six monolayers and no intermixing [11]. On the other hand, a similar UHV-MBE ex- 

periment of Ag on Ni(100) have found negligible coherency [22] with the first layer of 

Ag tending to close-packing. However, this may result from the competition between the 

surface energy and strain energy; another study [157] found Ag could grow in an almost 

coherent fashion on Ni(111) up to six monolayers, but with the silver domains rotated with 

respect to the substrate by ±2 ◦ .


Excess Volume 

There exists indirect evidence of excess volume 3 from elastic constant measurements. 

Carlotti et al. [31, 33, 32] found evidence of this in Ag/Ni multilayers, observing a reduc- 

tion of the effective shear elastic constant C44 by 40% when the bilayer thickness decreases 

from 18 to 2 nm using Brillouin light scattering. These results agree with the simulations of 

Jaszczak et al. [113], who suggested that excess volume at the interface can induce a dra- 

matic softening of C44. This “softening” is not seen in multilayers in which the components 

are mutually soluble. 

According to Jaszczak et al. [113], the excess volume at semicoherent boundaries con- 

tracts the in-plane lattice parameter by the Poisson effect; this in turn leads to an expansion 

in the Ni and Ag planes adjacent to the interfaces; these strain gradients were seen in Fig- 

ure 4.21. Furthermore, this expansion can explain the negative Poisson ratio found in these 

multilayers. Ni suffers a large expansion in the out-of-plane lattice parameter near the 

Ag/Ni interface. This expansion dominates over the Poisson contraction from the in-plane 

tensile stress. 4 

Implicit in the argument of Clemens et al. [39] is that the excess volume, �V , is zero 

[119] or at least does not affect the out-of-plane d-spacings near the interfaces. This clearly 

does not agree with our results. 

An excellent question is how the excess volume relates to the interface energy and 

3 Pelosin et al. [182, 183] have performed several annealing experiments that also indirectly show the 

presence of excess volume in Ag/Ni multilayers. The measured thermal expansion coefficient of Ag/Ni 

(19.1×10 −6 / ◦ C) decreases dramatically to zero when the sample is heated from room temperature to 375 ◦ C. 

This somewhat anomalous behavior can be explained by densification at the interfaces during annealing. As a 

side note, they measured a biaxial modulus of 200 GPa [182], a Young modulus of 142 GPa [184], and noted 

very little dislocation plasticity in the multilayers [182]. 

4 Other researchers have found negative Poisson ratios in Cu for Nb/Cu multilayers [59]. Nb/Cu also is 

similar to Ag/Ni in that it has large mismatch and low mutual solubilities. It is expected that a large excess 

volume also exists in Nb/Cu multilayers.


stress. Through simulations, Wolf has shown that there exists an almost linear relationship 

between excess volume and the interface energy [258, 259]. One may then expect that a 

compressive interface stress can expand the interface plane, and by a Poisson effect, reduce 

the excess volume and hence energy. Equation (4.21) shows one relationship between 

excess volume and interface stress, but further work along these lines may able to show a 

more direct relationship for a biaxial stress state. However, there is the added complication 

of the more coherent Ni/Ag interface affecting the Gibbs-Duhem relationship. 

Finally, an alternate approach to understanding the relationship between excess volume 

and interface stress can be made using a dislocation approach rather than a thermodynamic 

approach. 

Misfit Dislocations 

If we discount vacancy formation at the interfaces, the most significant contribution to 

excess volume at the interface is the misfit dislocation array. However, for a dislocation 

to have positive excess volume, anharmonic energy potentials are required [208]. We have 

already discussed the importance of anharmonic effects on the Matthews-Blakeslee crite- 

rion; this should be expected since the dislocations represent a singularity in the stress field. 

Near the cores, the strains no longer obey Hookean elasticity. 

If we take the interface energy to be positive, the problem of compressive interface 

stress can be reduced to the consideration of why 

∂γc 

∂ɛ 

+ ∂γs 

∂ɛ 

< 0. (4.38) 

While the first term is not necessarily small, we shall assume the second term dominates. 

Further work needs to done to verify this assumption.


Following Cammarata et al. [30], we can estimate the contribution of the higher order 

terms in the potential energy function on the interface stress, first by considering a one- 

dimensional array of dislocation cores, and then by using Matthews’ model of an edge 

dislocation array. 

Cammarata et al. [30] first assume a one term expansion of the harmonic potential: 

σ = Mcompɛ + M ′ comp ɛ2 . (4.39) 

The nonlinear contribution to the strain is equated to the volume expansion per unit length 

of a dislocation, which is approximately b 2 [208]. After determining the total strain energy 

per unit volume of a dislocation line, they find 

� � 

δ − εcoh 

fcore ≈−b 

2 

or with ξ ≈ (δ − εcoh)/2 (cf. equation (4.4)), 

fcore ≈− 

Mcomp 

� Mcomp 

M ′ comp 

� 2 

� Mcomp 

M ′ comp 

� 2 

(4.40) 

γ. (4.41) 

The Matthews model of interface energy is given by equation (4.4). We assume a 

completely relaxed interface such that εcoh = 0. Using the Shuttleworth relationship (4.19), 

we take the derivative of γ with respect to the interfacial strain, ɛ11, and obtain 

� 

f = bξ Mcomp 1 + 

ξ ′ 

ξ + M′ comp 

+ 

Mcomp 

b′ 

� 

b 

(4.42) 

where ( ′ ) represents the derivative. After considering the major terms using a Born-Mayer 

potential, Cammarata et al. find that 

� � 

M ′ 

comp 

f ≈ γ. (4.43) 

Mcomp


The higher order composite modulus, M ′ comp , is negative; hence the interface stress is neg- 

ative for a positive γ . Although the core contribution from equation (4.41) can be small, it 

also results in a compressive interface stress. 


The negative value of the Ag/Ni interface stress has been confirmed using the bulk Ag 

and Ni lattice parameters and is in agreement with previous results. Fits to low and high- 

angle x-ray scattering data show that intermixing is unlikely to affect the measurement of 

the Ag/Ni interface stress. The high-angle results also demonstrate the existence of excess 

volume at the Ag/Ni interface that causes an out-of-plane expansion in the adjacent layers 

and can result in a negative Poisson ratio. The excess volume originates from the volume 

expansion of dislocation cores, thereby emphasizing the contribution of anharmonic effects 

to the interface stress. 

Another way of examining asymmetric interdiffusion was demonstrated by Huai et al. 

[105]. They analyzed the low and high-angle x-ray scans of Co/Re and Re/Co bilayers. Do- 

ing similar experiments with Ag and Ni may be of interest. Also, experiments to measure 

the interface stress after a low temperature heat treatment (to avoid multilayer destratifica- 

tion [84, 197, 201, 199]) may sharpen the interfaces and refine the measurement. 

The added complications of interface roughness effects [212, 180] and grain boundary 

stresses [212, 16] on the interface stress are beginning to be studied, but the most important 

work that remains is a more rigorous calculation of the effect of an anharmonic potential on 

the interface stress, possibly considering the effects of core delocalization and dislocation 

interactions across the layer thickness.

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Appendix A 

X-Ray Stress Analysis 

Although the principle behind x-ray stress analysis (XSA) is simple, the actual exper- 

iment and analysis can be difficult. This section discusses some of the methods used to 

convert x-ray data into a stress tensor. 

By x-ray diffraction we directly measure 

dhkl = nλx−ray 

2 sin θhkl 

(A.1) 

where 2θhkl is the peak position of the (hkl)-reflection. We calculate the strain perpendic- 

ular to a particular set of (hkl) lattice planes by 

ɛ L 33 (hkl) = dhkl − dhkl, ref 

. (A.2) 

dhkl, 0 

The superscript L refers to the laboratory reference frame; the relationship between this 

coordinate system and the sample coordinate system, S, is shown in Figure A.1. The 

individual crystals have their own coordinate system denoted by C. Naturally, there are 

several ways to tilt by ψ; the most common is to rotate the sample normal in the plane of 

the diffractometer, i.e., the plane defined by the incoming and outgoing x-rays; since this 

145

Appendix A: X-Ray Stress Analysis 146 

S 1 

S 3 

ψ 

φ 

q, L3 Figure A.1: Diagram showing the relationship between the sample coordinate system, S, 

and the laboratory coordinate system, L. 

motor is often labeled ω, this is called the ω-diffractometer. Some diffractometers can tilt 

the sample normal in a plane orthogonal to the diffractometer plane; this is angle is usually 

labeled ψ in the literature, and so this is called the ψ-diffractometer. Some diffractometers, 

such as four-circle diffractometers, can tilt in both planes. In an attempt to avoid confusion, 

I will call angle in the plane orthogonal to the diffractometer plane χ and leave ψ as simply 

a general tilt angle. The angle φ is always called φ and refers to a rotation around the sample 

normal. 

It is best to analyze (hkl) reflections at high 2θ angles for the best sensitivity. Differen- 

tiating Bragg’s law gives 

S 2 

�θhkl = ɛ L 33 (hkl) tan θhkl 

(A.3) 

so for a given strain, �2θhkl is larger for higher 2θhkl. 

The measured strain is related to the strains in the sample reference frame by the rota-


tion matrix a(ψ, φ): 

ɛ L 33 (φ, ψ) = aLS 3k (ψ)aLS 3l (φ)ɛS kl 

= ɛ S 11 cos2 φ sin 2 ψ + ɛ S 22 sin2 φ sin 2 ψ + ɛ S 33 cos2 ψ + 

ɛ S 12 sin 2φ sin2 ψ + ɛ S 13 cos φ sin 2ψ + ɛS 23 sin φ sin 2ψ. (A.4) 

Therefore, by making six independent measurements of ɛ L 33 (φ, ψ), we can solve for 

all six components of ɛ S ij . Typically, however, we hope a plot of dhkl vs. sin 2 ψ yields a 

straight line; in such a case, ɛ S 13 = ɛS 23 

= 0. 

The stress tensor can then be determined using Hooke’s law, 

ɛ S ij = s S ijkl 

A.1 Untextured Thin Films 

S 

σkl . (A.5) 

Combining (A.4) and (A.5), we obtain for an elastically isotropic sample 

ɛ L 1 

33 (φ, ψ) = 

where S hkl 

i 

2 Shkl 

2 

+ 1 

2 Shkl 

2 

� 

σ S 11 cos2 φ sin 2 ψ + σ S 22 sin2 φ sin 2 ψ + σ S 33 cos2 � 

ψ + 

� 

σ S 12 sin 2φ sin2 ψ + σ S 13 cos φ sin 2ψ + σ S 23 sin φ sin 2ψ 

� 

+ 

S hkl 

� 

1 σ S 11 + σ S 22 + σ S � 

33 

are the x-ray elastic constants (XECs) for elastically isotropic materials. 

(A.6)


A.1.1 Biaxial Stress Case 

If we assume the stress tensor is biaxial, i.e., σi3 = 0, we can simplify this further and 

arrive at 

where 

Equibiaxial Stress Case 

ɛ L 33 (φ, ψ, hkl) = Shkl 

1 (σ S 11 + σ S 1 

22 ) + 

2 Shkl 

2 σ S φ sin2 ψ (A.7) 

σ S φ = σ S 11 cos2 φ + σ S 22 sin2 φ + σ S 12 sin 2φ. (A.8) 

Furthermore, in many cases σ S 11 = σ S 22 = σ S � and σ S 12 

stress state), and we can reduce (A.7) to 

ɛ L 33 (ψ, hkl) = 2Shkl 

1 σ S � 

Therefore, if S hkl 

1 and S hkl 

2 are known, the stress σ S � 

= 0 (resulting in an equibiaxial 

+ 1 

2 Shkl 

2 σ S � sin2 ψ. (A.9) 

can be determined from the slope of 

the dhkl(ψ) vs. sin 2 ψ line. Equation (A.9) is the commonly used equation for randomly 

oriented polycrystalline thin films. 

For an isotropic solid, we can also define S mech 

1 

and Smech 2 , which are independent 

of (hkl) and depend only on the macroscopic elastic properties of the material (and thus 

labelled the mechanical elastic constants): 

S mech 

1 

1 

2 Smech 2 = 1 + ν 

= −ν 

E 

(A.10) 

. (A.11) 

E


Another advantage of using S mech 

1 

and S mech 

2 

rather than S hkl 

1 and S hkl 

2 is that the XECs 

have no specific bounds while the mechanical elastic constants are bounded by the Voigt 

and Reuss limits or the Hashin-Shtrikman bounds [93, 94]. Kamminga et al. [123] have 

used formulas relating S hkl 

i 

σ S � 

to S mech 

i 

in a sample with known Smech 

1 

ɛ L 33 (ψ, hkl) = dhkl(ψ)(h 2 + k 2 + l 2 ) 1/2 − a0 

where 

= 

� 

2S mech 

1 + 2K2 

The values a0, K2, and σ S � 

aref 

� 

Ɣ − 1 

5 

for elastically isotropic samples so one can solve for 

and Smech 2 . Their equation is 

� 

+ 

� 

1 

2 Smech 

� 

2 − 3K2 Ɣ − 1 

�� 

sin 

5 

2 � 

ψ σ S � (A.12) 

Ɣ = h2k 2 + k2l 2 + l2h 2 

� 

h2 + k2 + l2 � . (A.13) 

2 

can then be determined by at least three measurements of 

dhkl(ψ)(h 2 + k 2 + l 2 ) 1/2 and performing a multivariate fit. This is called the direction 

solution method (or the Kamminga method) and the method I use for untextured thin films. 

A.2 Textured Thin Films in Equibiaxial Stress 

A major problem in stress analysis of textured thin films is that sufficient diffracted 

intensity can be found only at certain tilt angles, ψ. We can label these peaks as texture 

poles. For instance, if our sample has pure 〈111〉 fiber texture and we examine a (420) 

peak, we should tilt to 

� 

[111] · [420] 

ψ = arccos √ 

12 + 12 + 12 √ 42 + 22 + 02 � 

(A.14) 

to analyze strains in the 〈111〉-textured grains. Therefore, some have proposed only taking 

diffraction data at these texture poles (this is called the crystallite group method or CGM).


1 While this is useful for examining stresses in different grain populations (as Baker et al. 

[10] have done), extracting the macroscopic stress value in this manner can be dangerous 

if the degree of preferred orientation is not well known. For the situation of perfect 〈111〉 

fiber texture [123], the CGM gives 

where s C 0 = sC 11 − sC 12 

ɛ L 33 (ψ) = 

− 1 

2 sC 44 

, and 

ɛ L 33 (ψ) = 

for perfect 〈100〉 fiber texture [82]. 

� 

2s C 2 

12 + 

3 sC 0 

� 

2s C 12 + 

1 

+ 

2 sC 44 sin2 � 

ψ σ S � 

� 

1 

2 sC 44 + sC � 

0 sin 2 � 

ψ σ S � 

(A.15) 

(A.16) 

Therefore, after plotting d vs. sin 2 ψ for a sample with perfect 〈111〉 texture, the biaxial 

stress can be determined by 

∂ɛ L 33 (ψ) 

∂ sin2 1 

= 

ψ d0 

Additionally, the strain at ψ = 0isgivenby 

ɛ L 33 (ψ = 0) = 

∂d 

∂ sin2 ψ = sC 44 

2 σ S � . (A.17) 

� 

2s C 2 

12 + 

3 sC 0 

� 

σ S � . (A.18) 

We can use either equation to arrive at the stress or both to provide a check. 

A.3 The Strain-Free Lattice Parameter in Biaxially Stressed 

Samples 

For untextured films, the best way of obtaining a0 is by the direct solution method as 

mentioned earlier. The most common method of finding the strain-free lattice spacing for a 

1 Another way around the thin film intensity problem is to keep a large “footprint” on the sample by fixing 

the incoming beam at a low angle, such as 3 ◦ and to vary the tilt by taking measurements at different 2θhkls. 

This is known as the LIBAD method [227]. Of course in such a case, d is no longer linear in sin 2 ψ.


sample with known elastic constants is to solve (A.9) for zero strain, which gives 

sin 2 ψ0 = −4Shkl 

1 

S hkl 

2 

≈ −4Smech 

1 

Smech 2 

= 2ν 

. (A.19) 

1 + ν 

For a perfectly textured 〈111〉 sample, we can solve (A.15) for zero strain: 

sin 2 ψ0 = 2sC 44 − 4sC 11 − 8sC 12 

3sC . (A.20) 

44 

For more on x-ray stress analysis, including problems associated with triaxial stress 

states, see Noyan and Cohen [176] and Hauk [95].

Appendix B 

X-Ray Errors 

In this section, I assume the diffractometer is well aligned. Even after good alignment, 

however, we often have to contend with defocusing and geometric errors. In general, these 

errors can be accounted for by spreading a thin layer of strain-free powder onto the sample 

and measuring the peak shifts in the powder in addition to the peak shifts in the film. This 

may lead to diminished intensity from the sample and peak overlap, however. The second 

best option is to mount a strain-free powder sample and carry out the same measurements 

as for the sample of interest. The only remaining error here is the sample missetting; we 

will learn how to minimize this error. 

If either of these procedures are performed, the data should be reliable, and none of the 

corrections outlined in this appendix are required. However, I would advise those interested 

in XSA at least to skim this section to understand what the major sources of error are and 

how best to avoid them. The main lessons from this section are to minimize �X (the 

sample missetting from the 2θ-axis), to work at high 2θ, and to work at +ψ tilt angles 

when using the ω-diffractometer. 

152

Appendix B: X-Ray Errors 153 

We must define our angles carefully. ψ is, in general, the tilt of the sample normal with 

respect to the scattering vector. When working with an ω-diffractometer, i.e., working in 

ω-mode, ψ = �ω. When working in ψ-mode, ψ = χ. (�χ is unnecessary here because 

χ = 0 implies that the sample normal lies in the diffractometer plane). Also, I occasionally 

refer to η, the angle of rotation of the sample normal around the scattering vector [78]. η 

can be described in terms of �ω and χ but will be used only as a “switch” between ω-mode 

and ψ-mode: η = π for the ω-mode and π/2 for the ψ-mode. 

B.1 Mandatory Corrections 

B.1.1 Lorenz-Polarization-Absorption Corrections to the Intensity 

The LPA corrections are angle-dependent modifications to the intensity. One should 

divide the measured intensity by the following expressions: 

PR 

� �� 

1 + cos 2 2θ| cos 2θm| 

sin 2 θ 

� �� 

LR 

for an imperfect monochromator or 

PR 

� �� 

1 + cos 2 2θ cos 2 2θm 

sin 2 θ 

� �� 

LR 

(1 + tan ψ cot θ cos η) 

� �� 

AR 

(1 + tan ψ cot θ cos η) 

� �� 

AR 

(B.1) 

(B.2) 

for a perfect monochromator. Both these expressions refer to a “reflection” diffraction 

geometry rather than transmission. 

For a specimen of non-zero thickness (such as a thin film), the absorption correction is


different. In such instances, we multiply the above AR by 

� � 

1 − exp −µt 

1 

sin θ cos ψ − cos θ sin ψ cos η + 

�� 

1 

sin θ cos ψ + cos θ sin ψ cos η 

(B.3) 

where t is the film thickness and µ is the linear absorption coefficient. This can be sig- 

nificant for very thin films. For thin films in transmission, the total absorption correction 

is 

AT = 

exp(−µt/ cos αout) 

µ cos αin 

� 

1 1 

cos − αin cos αout 

where αin and αout are the incoming and outgoing angles. 

B.1.2 Refraction Corrections 

� 

� 

1 − e −µt 

� 

1 

cos α − 

in 1 

�� 

cos αout 

(B.4) 

Due to refraction, there is a correction to the 2θ position when working at low incidence 

angles 1 [225]: 

2θcorrect = (2θmeasured − αin) + 1 √ 2 

�� α 2 in − θ 2 �2 c + 4β 2 

�1/2 − θ 2 c + α2 �1/2 in 

(B.5) 

where β = (λx−rayµ)/(4π) is the imaginary part of the index of refraction, and θc is 

the critical angle. The values of β and θc can be found on the CXRO homepage, www- 

cxro.lbl.gov. There is a similar correction for grazing exit mode: 2 

2θcorrect = (2θmeasured − αout) + 1 √ 2 

where αout is the exit angle. 

�� α 2 out + θ 2 �2 c + 4β 2 

�1/2 + θ 2 c + α2 �1/2 out 

1 This correction is not needed for grazing incidence diffraction [148]. 

2 Caveat lector: I derived this formula myself. 

(B.6)


B.2 Geometric Errors 

B.2.1 Receiving Position Errors 

Following Wilson [251, 252], we define the error �2θ by 

2θcorrect = 2θ + �2θ, (B.7) 

where for defocusing errors associated with the receiving position of the detector, 

�2θdefocus = δr + δfr + δrs 

. (B.8) 

sin 2ϕ 

The terms δr + δfr + δrs have been made explicit by James and Cohen [112]. They refer to 

errors associated with the receiving position, correlated errors from the receiving position 

and x-ray focal point, and correlated errors from the receiving position and sample position, 

respectively. The average error due to these is given by ≈ 1/3 of 

�2θdefocus = R −2 

sd (Rsd 〈yr〉−〈xr〉〈yr〉+ 〈y2 r 〉 

cot 2θ (B.9) 

2 

−〈xr xs〉 cos(2θ − ψw) 

−〈yr ys〉 cos(2θ − ψw) 

−〈xr ys〉 sin(2θ − ψw) 

+ 〈yr xs〉 

sin 2θ 

� 

cos(4θ − ψw) + Rsd 

+ Rsd 

〈yr yf〉 cot 2θ) 

Rxs 

where I have assumed 〈xr〉〈yr〉 =〈xryr〉, 

Rxs is the x-ray focus to sample distance, 

Rsd is the sample to detector distance, 

Rxs 

� 

cos ψw cos 2θ


ψ = ∆ω 

O 

ψ = 0° 

Rxs 

Rsd 

Figure B.1: The focusing circle. 

xf and yf are the deviations of the x-ray focus from point A, 

xs and ys are the deviations of the sample surface from O, 

xr and yr are the deviations of the PSD from B 


ψw = θ + �ω. 

See Figure B.1 for A, B, and O. It may be obvious from the formulation that a good 

estimate of this error is difficult to obtain. The formulation is good from a heuristic point 

of view, however; maximizing Rsd and the 2θ angle reduces defocusing errors. 

B 

q 

A


B.2.2 Horizontal Divergence Errors 

Since the incoming beam has finite divergence, the maximum in diffracted intensity 

varies with the sample-detector geometry. The peak shift in degrees (for a flat plate sample) 

is given by ≈ 1/3of 

�2θdiv = 360 

π 

Rxs 

Rsd 

tan 

� δ 

2 

⎛ 

� 

⎜ 

sin(θ − �ω + δ/2) sin(θ − �ω − δ/2) ⎟ 

⎝ 

− ⎟ 

sin(θ + �ω − δ/2) sin(θ + �ω + δ/2) ⎠ . 

� �� 

(B.10) 

BO 

AO 

If the reader is using Noyan and Cohen [176], �ω + θ − δ/2 = �NC and π − �ω − 

θ − δ/2 = χNC, and the two terms in the parentheses are their BO and AO, respectively. 

Note that their error is much smaller since they move the detector to an optimized position. 

Horizontal errors can be significant if using small incidence angles; again, a strain-free 

powder should be used to correct for this error. 

B.2.3 Alignment Errors 

Specimen and Beam Displacement 

For the ω-mode, the error in 2θ due to specimen and beam displacement is given by 

�2θω−disp ≈ 180 sin θ(2�Xω cos θ − �eω sin(θ − �ω)) 

π 

Rsd sin(θ + �ω) 

⎞ 

(B.11) 

where �Xω and �eω are the missettings of the specimen surface and incident beam with 

respect to the 2θ-rotation axis, respectively. This effect is largest for low Bragg angles. The 

specimen displacement, �Xω, also affects the calibration constant � (described in section 

B.3.3), and is extremely important to minimize. This error is negligible for systems with 

both incoming and outgoing parallel x-rays [176].


The resulting 2θ shift (�2θ�ω − �2θ0) due to these errors is given by [41] 

δ(�2θ) = 180 

� 

2 cos θ sin θ sin �ω 

π Rsd sin(�ω + θ) �eω 

� 

� � 

sin θ 

− 1 − 

�Xω . (B.12) 

sin(θ + �ω) 

Assuming the beam displacement is negligible, Cohen [40] developed an easy way to 

check the sample displacement for cubic materials using the equation: 

ahkl − a0 

a0 

=− �Xω 

Rsd 

cos2 θ 

. (B.13) 


By determining the lattice parameter ahkl for various reflections at �ω = 0 and plotting 

ahkl vs. cos 2 θ/sin θ, the fitted slope is equal to �Xωa0/Rsd. a0 is given by the intercept. 

A positive slope indicates that the sample is displaced to a position behind the center of 

the diffractometer. I often use this method for FCC thin films with the (111) and (222) 

reflections. 

For materials with low stacking fault energies, the Cohen method of eliminating sample 

displacement may not work. For such cases, see Noyan and Cohen [176]. 

For the ψ-mode, the error due to sample and beam displacement off the ψ-rotation axis 

is given by: 

�2θψ−disp = 180 

π 

2 

Rsd 

cos θ � � 

�Xψ − sin χ�eψ 

cos χ 

(B.14) 

where �eψ is the missetting of the incident beam with the ψ-rotation axis and �Xψ is the 

missetting of the sample surface with respect to the ψ-rotation axis. 

The shift �2θχ − �2θ0 is given by [41]: 

δ(�2θ) = 180 

π 

2 

Rsd 

cos θ 

�� 

1 − 1 

� 

� 

�Xψ + �eψ tan χ 

cos χ 

(B.15) 

Generally, we should use low θ and high χ to measure �eψ and either eliminate or account 

for it.


Effect of the ω-Axis Not Corresponding to the 2θ-Axis 

For the ω-diffractometer, this offset gives an error 

where X ′ is the offset. 

�2θω−axis = 180 (�X 

π 

′ ) sin 2θ(1 − cos �ω) 


δ(�2θ) = 180 (�X 

π 

′ ) sin 2θ(1 − cos �ω) 


(B.16) 

(B.17) 

In general, this error should be negligible if the Eulerian cradle is mounted properly 

onto the two-circle base. This mounting should be performed at a machine shop using a 

dial counter. This was not done for the Harvard four-circle, so �X ′ ≈ 25 µm. 

B.3 Corrections for a Position Sensitive Detector 

B.3.1 Parallax Corrections for a PSD 

or 

According to James and Cohen [112], the true value of 2θ is given by 

2θ = 2θPSD + arctan(n tan �) (B.18) 

� � 

nz 

2θ = 2θPSD + arctan 

Rsd 

(B.19) 

where 2θPSD is the 2θ value at which lies the center channel of the PSD, n is the number 

of pixels away from 2θPSD, z is the length per pixel and � is the number of degrees 2θ per 

pixel for the PSD.


B.3.2 Small Beam Spot Corrections 

When using a PSD, one needs to oscillate the sample (usually around ω) by a couple of 

degrees in order to sample enough grains for good statistics [111]. 

B.3.3 PSD Calibration Constant Errors 

The calibration constants the position along the PSD into degrees 2θ and depends on 

the sample to detector distance; if there is non-zero �Xω, the calibration will be off. The 

calibration constant is given by 

� = 

second peak − peak at center channel (in degrees) 

channel difference 

(B.20) 

where �(Rsd) is determined for a fixed Rsd using a calibration sample. If the next sample 

is misset by �Xω, the calculated peak shift will be error due to an incorrect �(Rsd) given 

by 

σcal = �2θ �(Rsd) − �(Rsd + �Xω) 

. (B.21) 

�(Rsd) 

In general, I ignore this correction by ensuring that �Xω is small. 

B.4 Counting and Fitting Errors 

The variance in d as a function of variance in the 2θ peak position is given by [255] 

σ 2 (d) = 

where σ(2θ) is given in degrees. 

� �2 λ cos θp σ 2 (2θp) 

2 sin 2 θp 

4 

� 

π 

�2 180 

(B.22)

Appendix C 

Alignment Procedure for the Harvard 

Four-Circle Diffractometer 

In general, alignment consists of three major steps: 

• alignment of the diffractometer with itself, 

• alignment of the x-rays with the diffractometer (2θ)-axis, and 

• alignment of the sample surface with the 2θ-axis. 

The first step refers to ensuring the Eulerian cradle is properly mounted onto the two-circle 

base. We focus on the latter two steps. 

C.1 Coarse Alignment 

We can begin with Geremia’s method [81]. This involves first aligning a pinhole colli- 

mator in place of the sample and then maximizing the intensity of the beam coming through 

161

Appendix C: Alignment Procedure for the Harvard Four-Circle Diffractometer 162 

the pinhole. 

C.2 Fine Alignment 

Good studies on the elimination of both sample and x-ray tube missettings were com- 

pleted by Convert and Miege [41] and Vermeulen and Houtman [235]. This section is based 

primarily on their work. 

Aligning the x-rays with the diffractometer axis can be performed by mounting a stan- 

dard into the diffractometer (such as NIST’s Si powder 640) and performing the following 

procedure. 

We refer back to an equation in the previous section: 

δ(�2θ) = 180 2 cos θ 

π Rsd 

� sin θ sin �ω 

sin(�ω + θ) �eω − 

� 

1 − 

� � 


�Xω 

sin(θ + �ω) 

(C.1) 

where �eω is the displacement of the x-ray tube from the diffractometer axis and �Xω is 

the displacement of the sample surface from the diffractometer axis. The goal here is to 

move the x-ray tube in the horizontal plane until �eω = 0. One can do this even when 

there exists a finite �Xω by tilting the sample by an amount �ωc where 

1 − 

the solution is simply (besides �ω = 0) 


sin(θ + �ωc) 

= 0; (C.2) 

�ωc = π − 2θ. (C.3) 

When �ω = �ωc, the shift in 2θ is zero when �eω = 0. �Xω can be zeroed afterwards. 

Unfortunately, there is no similar method with �eψ and the equation 

δ(�2θ) = 180 

π 

2 

�� 

cos θ 1 − 1 

� 

� 

�Xψ + �eψ tan χ . 

cos χ 

(C.4) 

Rsd


We have to determine �eψ and �Xψ by fitting to the diffraction results for an unstrained 

sample to this equation. 

C.3 Aligning Samples 

The first thing to do after mounting a sample is to ensure that the sample surface normal 

lies in the diffraction plane by using the laser. Reflect the laser off the surface and mark its 

position on a backdrop. Rotate the sample around its normal by 180 ◦ and again mark the 

reflected spot. The midpoint between the two spots is the ideal location. Adjust the sample 

goniometer accordingly. 

Once �eω and �eψ are zero, the most important adjustment is to minimize �Xω. This 

can be done with Cohen’s method, as discussed in the previous section. Many thin films 

are textured, however, and only one or two peaks are available (or the two peaks differ 

in strain). As stated in Appendix B, usually the (111) and (222) reflections can be used. 

Also, we can take advantage of films with smooth surfaces and align the sample by x-ray 

reflectivity. Move the sample in the diffraction plane until the surface splits the main beam 

in half. Then rotate the detector to 1 ◦ and adjust ω (or whatever rotates the sample normal in 

the diffraction plane) such that you obtain a reflection peak at 1 ◦ . Then set ω to 0.5 ◦ . Bring 

the detector back to 0 ◦ and split the beam in half again. Keep reiterating this procedure 

until everything agrees.


C.4 Calibrating the Position Sensitive Detector 

First, make sure the x-ray tube is aligned well. The PSD is then calibrated by the 

following routine. 

1. Guess the starting parameters for the PSD. Right now, the center channel of the PSD 

is ≈ 868, tan � = 0.00027, z = 0.04003 mm/chn, Rxs = 147 mm, and Rsd = 

147.702 mm. 

2. Mount and align the powder standard so that �X = 0. 

3. Choose a detector position featuring two high 2θhkl peaks, one at the center channel 

and another off to the left or right side of the PSD. Adjust the detector so that one 

of the diffraction peaks lies at the center channel; record this scan. Then pull the 

detector away from the sample and perform a distant scan. Finally, push the detector 

back into the previous position for a scan; the peak should stay at the same channel 

for both the far and the near scans! If it doesn’t, adjust the slide on which the de- 

tector sits until the detector slide is aligned with path of the diffracted x-rays. This 

determines the true “center channel”. 

4. Now that we know the sample and x-ray tube are perfectly adjusted, and now that 

we know the precise value of the center channel, we can use this to obtain the exact 

values of z and Rsd. Wefirst ensure the same diffraction peak as before lies on the 

center channel. Then we pull the detector out by a known amount and take a scan; 

this is followed by pushing the detector in, and taking a scan. We can use this data 

and value of astandard to obtain z and Rsd.

Appendix D 

Elastic Constants 

Throughout this appendix, we will use the prime ( ′ ) to denote quantities in the sample 

coordinate system and no prime to denote quantities in the crystal coordinate system. 

D.1 Important Elastic Quantities 

For cubic single crystals, the Young modulus (E), Poisson ratio (ν), and biaxial modu- 

lus (Y ) are defined by: 

E = (c′ 11 − c′ 12 )(c′ 11 + 2c′ 12 ) 

c ′ 11 + c′ 12 

ν = 

c ′ 12 

c ′ 11 + c′ 12 

Y = c ′ 11 + c′ 12 − 2c2 13 

c ′ 33 

165 

= −s′ 12 

s ′ 11 

= 

= 1 

s ′ 11 

1 

s ′ 11 + s′ 12 

(D.1) 

(D.2) 

. (D.3)

Appendix D: Elastic Constants 166 

D.2 The Rotation Matrix 

When transforming from the crystal coordinate system to the sample coordinate system, 

we use the Euler angles (ϕ1,�,ϕ2) where ϕ1 is the rotation around the [001] axis, � is the 

rotation around the new [100] ′ axis, and ϕ2 is the rotation around the new [001] ′′ axis. Such 

a rotation matrix is given by 

� 

a = 

cos ϕ1 cos ϕ2 − cos � sin ϕ1 sin ϕ2 cos ϕ2 sin ϕ1 + cos � cos ϕ1 sin ϕ2 

− cos � cos ϕ2 sin ϕ1 − cos ϕ1 sin ϕ2 cos � cos ϕ1 cos ϕ2 − sin ϕ1 sin ϕ2 

sin � sin ϕ1 

and any tensor quantity of order n can be transformed by 

− cos ϕ1 sin � 

sin � sin ϕ2 

� 

cos ϕ2 sin � 

cos � 

(D.4) 

M ′ = a n M. (D.5) 

One can then easily determine the following relationships for (111)-oriented crystals 

(ϕ1 =−π/4,�=−arccos(1/ √ 3), ϕ2 = 0) 

c ′ 11 

c ′ 33 

c ′ 12 

c ′ 13 

= 1 

2 (c11 + c12 + 2c44) (D.6) 

= 1 

3 (c11 + 2c12 + 4c44) (D.7) 

= 1 

6 (c11 + 5c12 − 2c44) (D.8) 

= 1 

3 (c11 + 2c12 − 2c44) (D.9)


s ′ 12 

s ′ 11 

= 1 

4 (2s11 + 2s12 + s44) (D.10) 

= 1 

12 (2s11 + 10s12 − s44) . (D.11) 

D.3 Average Elastic Quantities For Polycrystalline Aggre- 

gates 

In general, a tensor property can be averaged by integrating over all space using the 

Euler angles and normalizing. For example, a fourth order tensor Mijkl can be averaged 

using the equation 

〈Mijkl〉= 1 

8π 2 

� ϕ1=2π � �=π � ϕ2=2π 

Mijkl(ϕ1,�,ϕ2) sin � dϕ1 d� dϕ2. (D.12) 

ϕ1=0 �=0 ϕ2=0 

With cubic symmetry, this simplifies to 

〈Mijkl〉= 4 

π 2 

� ϕ1=π/2 � �=π/2 � ϕ2=π/2 

ϕ1=0 

�=0 

ϕ2=0 

Mijkl(ϕ1,�,ϕ2) sin � dϕ1 d� dϕ2 (D.13) 

since all the information is contained within an octant in space. If the material is textured, 

an orientation distribution function (ODF) f (ϕ1,�,ϕ2) (and its associated normalization 

constant) may be used as a weighting factor. 

D.4 The Vook-Witt Model of Grain Interaction 

If we take every grain in the film to have equal strain tensors (thus providing stress 

discontinuities at the grain boundaries) we can average over the cmn and obtain a Voigt


average. If we assume equal stress tensors in each grain (thus violating the strain com- 

patibility relations), we can average over the smn and obtain a Reuss average. These are 

called “grain interaction” models. Hill showed that the Voigt average is an upper bound 

while the Reuss average is a lower bound [100]. He thus proposed the Hill average for the 

elastic constants, 〈Mijkl〉=(1/2)(〈M Voigt 

ijkl 〉+〈MReuss 

ijkl 〉). Vook and Witt suggested a grain 

interaction model particularly suited to thin films with a columnar morphology, where each 

grain has the following strain elements [239]: 

ɛ ′ 11 = ɛ′ 22 

ɛ ′ 12 = ɛ′ 21 

σ ′ 

i3 = σ ′ 3i 

= e (D.14) 

= 0 (D.15) 

= 0. (D.16) 

The remaining strain and stress tensor elements can be solved for using Hooke’s law. They 

are given by 

where 

σ ′ 1 = 

σ ′ 2 = 

� 

e s ′ 2 

26 − s ′ 16s′ 26 + s′ � 

66 s ′ 

12 − s ′ � 

22 

� 

ζ 

� 

e s ′ 2 

16 − s ′ 16s′ 26 + s′ 66 

σ ′ 6 = e � s ′ 11s′ 26 + s′ 22s′ 16 − s′ 12 

2ζ 

ζ 

� 

s ′ 

12 − s ′ � 

11 

� 

� 

s ′ 

16 + s ′ �� 

26 

ζ = s ′ 2 

16 s ′ 22 − 2s′ 12 s′ 16 s′ 26 + s′ 11 s′ 2 

26 + s ′ 2 

12 s ′ 66 − s′ 11 s′ 22 s′ 66 

(D.17) 

(D.18) 

(D.19) 

(D.20)


and (using the above stresses) 

ɛ ′ i 

� 

= e 2s ′ 2 

16s ′ 2i − 2s′ 16s′ 26 (s′ 1i + s′ 2i ) − s′ 16s′ i6 (s′ 12 − s′ 22 ) + 

where i = 3, 4, 5. 

� 

/ 

� 

2(s ′ 2 

16s ′ 22 − 2s′ 12s′ 16s′ 26 + s′ 11s′ 2 

26 + s ′ 2 

12s ′ 66 − s′ 11s′ 22s′ 66 ) 

� 

2s ′ 1i (s′ 2 

26 + s ′ 12 s′ 66 − s′ 22 s′ 66 ) + (s′ 11 − s′ 12 )(s′ 26 s′ i6 − 2s′ 2i s′ 66 ) 

(D.21) 

The Vook-Witt model may have important consequences for x-ray diffraction work as 

well as for elastic parameters [229], but this remains to be corroborated. 

D.4.1 The Hornstra-Bartels Calculation Technique 

For the following elastic calculations, the method of Hornstra and Bartels [102, 202] 

has been used. This method speeds up the calculations considerably. According to this 

formulation (which relies on the Vook-Witt model), the strain in each grain is given by 

ɛij = eδij + aiℓj 

(D.22) 

where δij is the Krönecker delta, ℓ j is the interface normal (unit vector), and ai is a vector 

to be solved for. 

The strain elements are found by using equation (D.22). We can simplify the index


notation using the definitions 

ɛ1 = ɛ11 (D.23) 

ɛ2 = ɛ22 (D.24) 

ɛ3 = ɛ33 (D.25) 

ɛ4 = ɛ23 + ɛ32 (D.26) 

ɛ5 = ɛ13 + ɛ31 (D.27) 

ɛ6 = ɛ12 + ɛ21. (D.28) 

We can then write the stress elements as (again for cubic crystals) 

σ1 = c11ɛ1 + c12(ɛ2 + ɛ3) (D.29) 

σ2 = c11ɛ2 + c12(ɛ3 + ɛ1) (D.30) 

σ3 = c11ɛ3 + c12(ɛ1 + ɛ2) (D.31) 

σ4 = c44ɛ4 (D.32) 

σ5 = c44ɛ5 (D.33) 

σ6 = c44ɛ6. (D.34) 

The solution for ai can be obtained by using the above equations and again noting that the 

components of force normal to the sample are zero: 

σ1ℓ1 + σ6ℓ2 + σ5ℓ3 = 0 (D.35) 

σ6ℓ1 + σ2ℓ2 + σ4ℓ3 = 0 (D.36) 

σ5ℓ1 + σ4ℓ2 + σ3ℓ3 = 0. (D.37) 

This gives ai in terms of ℓ j, e, c11, c12, and c44. The stress and strain elements in the


sample coordinates are then given by 


σ ′ 

ij = aika jlσkl 

D.4.2 Results with the Vook-Witt Model 

(D.38) 

ɛ ′ ij = aika jlɛkl. (D.39) 

Using the Vook-Witt model and the Hornstra-Bartels calculation method, we can deter- 

mine some useful elastic properties for anisotropic materials. 

Following the example of Murakami and Chaudhari [166], I calculated the ratio of 

out-of-plane strain to in-plane strain (Figure D.1) and the biaxial modulus (Figure D.2) 

and mapped them onto a (111) stereographic projection (the (111) plane is parallel to the 

figure). 

For FCC materials such as Ni and Cu, slip preferentially occurs on the {111} planes in 

the 〈110〉 directions. The maximum resolved shear stress in a crystal with a surface normal 

[hkl] is shown in Figure D.3 for an arbitrary value of ɛ ′ 11 

= 0.704% (originally used to 

compare my results with those of Murakami and Chaudhari). This shows that orientations 

with the {331} and {110} family of planes parallel to the substrate yield before the {111} 

family, which in turn yield before the {100} family.



112 

Cu 

112 

101 313 

113 

101 313 

113 

211 

001 

211 

001 

111 

113 

111 

113 

123 

123 

110 

111 

110 

111 

010 

110 

010 

110 

131 

121 

131 

131 

121 

131 

111 

111 

Figure D.1: The ratio of ɛ ′ 33 /ɛ′ 11 in the Vook-Witt model. 

133 

133 

011 

011 

112 

112


112 

112 


101 313 


113 

101 313 

113 

211 

001 

211 

001 

111 

113 

111 

113 

123 

123 

Figure D.2: The biaxial modulus in units of 100 GPa using the Vook-Witt model. 

110 

111 

110 

111 

010 

110 

010 

110 

131 

121 

131 

131 

121 

131 

111 

111 

133 

133 

011 

011 

112 

112



112 


112 

101 313 

113 

101 313 

113 

211 

001 

211 

001 

111 

113 

111 

113 

133 

133 

011 

011 

Figure D.3: The maximum resolved shear stress on the (111) slip plane given in GPa using 

ɛ ′ 11 = 0.704%. 

110 

111 

110 

111 

010 

110 

010 

110 

131 

131 

121 

131 

121 

131 

111 

111 

133 

133 

011 

011 

112 

112

Appendix E 

The Stoney Equation 

We consider a thin film on a substrate, with the bottom layer of the substrate as the 

z = 0 reference plane [66]. The lattice parameter of the composite is allowed to vary such 

that the misfit strain ɛm 11 is given by 

From the strain compatibility equations, 

ɛ m 11 = d0 − d(z) 

. (E.1) 

d(z) 

ɛij(z) = ɛ m ij (z) − zκij(t) + ɛ r ij (t) (E.2) 

where κij(t) is the curvature of the reference plane at a total composite thickness t and 

ɛr ij (t) is the strain of the reference plane at thickness t. 

In what follows, we will focus on ɛ11(z) and therefore use the notation 

ɛ m 11 = ɛm 22 

ɛ r 11 = ɛr 22 

= ɛm 

= ɛr 

κ11 = κ22 = κ. 

175

Appendix E: The Stoney Equation 176 

If we assume the film has a constant misfit strain ɛm and the substrate has none, 


Then we use force and moment balance: 


The force balance term written out is 

Ys 

� ts 

and the moment balance term is 

Ys 

0 

� ts 

0 

ɛ s 11 (z) =−zκ + ɛr 

(E.3) 

ɛ f 11 (z) = ɛm − zκ + ɛr. (E.4) 

� tf+ts 

dzσ11(z) = 0 (E.5) 

0 

� tf+ts 

dzσ11(z)z = 0. (E.6) 

0 

dzɛ s 11 (z) + Yf 

dzɛ s 11 (z)z + Yf 

� ts+hf 

ts 

� ts+hf 

ts 

dzɛ f 11 (z) = 0 (E.7) 

dzɛ f 11 (z)z = 0. (E.8) 

Substituting in the above equations for ɛs 11 (z) and ɛf 11 (z) and integrating gives two equations 

with the two unknowns κ and ɛr. Solving for these gives 


ɛr = ξYftf 

κ = 6ξYfYstfts(tf + ts)ɛm 

� 

Yst 2 s (3tf + 2ts) − Yft 3 � 

f 

(E.9) 

(E.10)


where 

ξ = 

When we use σ f 11 = Yfɛm we see that 

� 

Y 2 

f t4 f + Y 2 s t4 s + 2YfYstfts(2t 2 f + 3tfts + 2t 2 s ) 

�−1 . (E.11) 

σ f 11 = 

κ 

. (E.12) 

6ξYstfts(tf + ts) 

Freund et al. [67] have a more attractive expression for equation (E.9): 

κ = 6ɛm 

ts 

where t = tf/ts and Y = Yf/Ys. 

� 

1 + t 

1 + tY(4 + 6t + 4t2 ) + t4Y 2 

� 

(E.13) 

Stoney’s approximation is used when tf ≪ ts (such that ξ → (Y 2 s t4 s )−1 ) so that this 

simplifies to 


σ f 11 = Yst 2 s 

6tf 

κ (E.14) 

ɛr = 2 Yftf 

. (E.15) 

Ysts 

Note that there is an implicit assumption in Stoney’s equation that the biaxial modulus 

of the film in near or less than that of the substrate; strictly speaking, ξ really goes to 

(Y 2 s t4 s + 4YfYstft 3 s )−1 . If Yf is much greater than Ys (near (tsYs)/(4tf) — which may be 

possible for rigid films on polymer substrates), then one should use the approximations 

σ f 11 = 

� 

Yst 2 s 

+ 

6tf 

2Yfts 

� 

κ (E.16) 

3 


ɛr = 

2Yftf 

. (E.17) 

Ysts + 4Yftf



When the film misfit strain ɛm is not constant through the thickness of the film, we take 

ɛ s 11 (z) =−zκ + ɛr 

(E.18) 

ɛ f 11 (z) = ɛm(z) − zκ + ɛr. (E.19) 

This gives for κ and ɛr, 

κ =−6Yfξ(Yst 2 s + Yftf(tf 

� ts+tf 

+ 2ts)) dzɛm(z) + 

ts 

� ts+tf 

12Yfξ(Yftf + Ysts) dzɛm(z)z (E.20) 

ts 


ɛr =−4Yfξ(Yst 3 s + Yftf(t 2 f + 3tfts + 3t 2 s )) 

� ts+tf 

dzɛm(z) + 

ts 

6Yfξ(Yst 2 s + Yftf(tf 

� ts+tf 

+ 2ts)) dzɛm(z)z. (E.21) 

In the thin film approximation, 

κ = 6Yf 

Yst 2 � � ts+tf 

� ts+tf 

2 

dzɛm(z)z − 

s ts ts 

ts 

or 

or in terms of force per unit width, 

ts 

� 

dzɛm(z) 

(E.22) 

ɛm(tf) = Yst 2 s 

κ 

6Yf 

′ (tf) (E.23) 

ɛm(tf) = 1 

Yf 

� F ′ (tf)/w � 

(E.24) 

where κ ′ (tf) and F ′ (tf)w −1 are the slopes of the curvature or force per unit width versus 

thickness curves, respectively. This allows an extraction of the strain profile from curvature 

data.

Appendix F 

List of Symbols 

179

Appendix F: List of Symbols 180 

A area 

a lattice spacing 

a0 equilibrium lattice spacing 

aref reference lattice spacing 

b Burgers vector 

cijkl elastic stiffness 

cos λ cos φ Schmid factor 

Dδ misfit dislocation spacing 

d grain size 

dα, dβ out-of-plane d-spacings of the α or β-phase 

E Young’s modulus 

Eact activation energy 

F force 

fij interface stress 

fhkl volume fraction of hkl-oriented grains 

G shear modulus or Gibbs free energy 

GID grazing incidence diffraction 

GND geometrically necessary dislocation 

gij relative interface stress 

Mcomp composite modulus of the film and substrate 

Nα, Nβ number of monolayers of the α or β-phase 

P hydrostatic pressure 

q scattering vector 

RBS Rutherford Backscattering Spectrometry 

SSD statistically stored dislocation 

sijkl elastic compliance 

Tij thermodynamic tensions 

T temperature or dislocation line tension 

t film thickness 

U energy 

ui displacement 

w work 

Y biaxial modulus 

YU unrelaxed biaxial modulus 

relaxed biaxial modulus 

YR

Appendix F: List of Symbols 181 

α coefficient of thermal expansion or incidence angle for x-ray scattering 

β dislocation cut-off parameter 

γ interface energy 

γc, γs chemical or structural components to the solid-solid interface energy 

�V excess volume 

δ misfit or the real part of the index of refraction 

ɛij strain 

ɛsh,ɛ�,ɛ⊥ shear strain, in-plane strain, and out-of-plane strain 

εcoh coherency strain 

ζ Ronay’s geometric factor 

ηijk 

Lagrangian strains 

θ diffraction angle/2 

κ curvature 

� bilayer thickness 

λ x-ray wavelength 

µ chemical potential 

ν Poisson’s ratio 

ξ dimensionless parameter for the stress field near a dislocation core 

ρ dislocation density 

σij stress 

σ shear stress, interface roughness, or standard deviation 

σy yield strength 

σflow flow strength 

σsc substrate curvature stress 

volume stress 

〈σ 〉v 

〈σ 〉x−ray 

x-ray measured volume stress 

τ characteristic relaxation time or 1/e penetration depth 

ϕ1,�,ϕ2 Euler angles 

χ sample tilt angle out of the diffractometer plane 

ψ general sample tilt angle used in the sin 2 ψ-method 

ω strain oscillation frequency 

ℓ characteristic length scale for plasticity

Stresses in Cu Thin Films and Ag/Ni Multilayers - Harvard School of ...

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