Diploma Thesis - Erich Schmid Institute

Thermal Behaviour of Al/Si(0 0 1) and GaN/BGaN/Al2O3(0 0 0 1)
structures characterized using X-ray diffraction
Diploma Thesis
submitted by
Hafok Martin
Erich Schmid Institut für Materialwissenschaft
Leoben, March 2004

Danksagung
Diese Arbeit entstand im Zeitraum September 2003 bis März 2004, wobei sämtliche
Messungen am Erich Schmid Institut für Materialwissenschaft in Leoben durchgeführt
wurden.
Das Schreiben dieser Arbeit wäre ohne die professionelle Hilfe und Unterstützung von
Herrn Dr. DI. Jozef Keckes nicht möglich gewesen. Ich verdanke ihm, dass er mir die
Methoden der Spannungsmessung mittels Röntgendiffraktion sowie deren
Auswertung und Interpretation näher gebracht hat. Ebenso möchte ich seinen Einsatz
und guten Rat, der mir bei der Lösung vieler Probleme im Rahmen dieser Arbeit
geholfen hat, hervorheben sowie dass er für mich und meine Anliegen immer Zeit
gefunden hat. Aus all diesen Gründen will ich mich herzlich bedanken.
Weiters möchte ich auch Herrn Ao. Univ. Prof. Balder Ortner meinen Dank für seine
Ratschläge und Bemühungen aussprechen.
Ein weiterer großer Dank gilt Herrn DI. Ernst Eiper, der mir bei der schwierigen
Messung der einkristallinen GaN/BGaN/Saphire Probe behilflich war und mir bei der
Korrektur der Arbeit eine wertvolle Hilfestellung gegeben hat. Ebenso möchte ich mich
auch bei Stefan Massl bedanken, der mir bei der Überarbeitung des Textes geholfen
hat.
Zu guter letzt möchte ich meiner Familie meinen Dank aussprechen, die mich in
meinem Bestreben stets unterstützt hat und mir immer einen wertvollen Rückhalt
geboten hat.

Content:
i
Content
1. Motivation.............................................................................................................1
2. Introduction ..........................................................................................................2
3. Basic X-ray diffraction expressions ......................................................................3
3.1. Reciprocal lattice ......................................................................................3
3.2. Bragg equation .........................................................................................4
4. Mechanical properties of crystals .........................................................................6
4.1. Stress, strain and displacement ...............................................................6
4.2. Anisotropic Elasticity.................................................................................7
4.3. Intrinsic and extrinsic stress .....................................................................9
5. Orthogonal tensors and rotation.........................................................................11
5.1. Basic relations ........................................................................................11
5.2. Rotation in a plane..................................................................................12
5.3. Euler angles ...........................................................................................13
6. Basic Physical Properties of layer and substrate materials ................................15
6.1. Polycrystalline aluminium on monocrystalline silicon .............................15
6.1.1. Properties of aluminium layer .................................................................15
6.1.2. Properties of silicon substrate ................................................................16
6.2. Monocrystalline GaN on monocrystalline sapphire.................................17
6.2.1. GaN........................................................................................................17
6.2.2. Boron nitride ...........................................................................................21
6.2.3. Sapphire substrate .................................................................................22
7. Polycrystalline and monocrystalline Model.........................................................24
7.1. Calculating isotropic elastic constants....................................................24
7.1.1. Isotropic material ....................................................................................24
7.1.2. Voigt model ............................................................................................24
7.1.3. Reuss model: .........................................................................................27
7.1.4. Hill model................................................................................................28
7.2. Calculating stresses for a single crystalline material ..............................29
7.3. Strain evaluation.....................................................................................31

ii
Content
8. Deposition of Thin Films.....................................................................................33
8.1. Magnetron sputtering of polycrystalline Al thin films on Si(1 0 0) ...........33
8.2. Molecular beam epitaxy of GaN/BGaN on Al2O3(0 0 0 1).......................34
9. X-ray Diffraction – Measurements and Alignment ..............................................36
9.1. Four Circle Goniometer ..........................................................................36
9.2. DHS 900 Domed Hot Stage ...................................................................37
9.3. The High-Resolution Monochromator.....................................................38
9.4. Alignment of the diffractometer – point focus .........................................41
9.5. Alignment of the diffractometer – line focus............................................42
10. Aluminium on silicon measurement....................................................................45
10.1. Experiment .............................................................................................45
10.2. Shift of diffraction peaks .........................................................................46
10.3. Peak broadening with increasing ψ tilt....................................................48
11. GaN and GaBN on sapphire measurement........................................................50
11.1. Measuring with the high resolution monochromator ...............................50
11.2. Stereographic projections and crystal orientation...................................51
11.3. Phi adjustment........................................................................................55
11.4. Omega adjustment .................................................................................57
11.5. Theta scans:...........................................................................................58
12. Results of aluminium on silicon ..........................................................................61
12.1. Sin(ψ)² vs. a plot.....................................................................................61
12.2. Lattice spacing of aluminium layer .........................................................64
12.3. Thermal expansion coefficient of aluminium...........................................65
12.4. Lattice spacing of silicon substrate.........................................................66
12.5. Thermal expansion coefficient of silicon substrate .................................66
12.6. Stress curve ...........................................................................................67
12.7. Strain curves ..........................................................................................69
12.8. Discussion ..............................................................................................70
13. Results of GaN/GaBN/Al2O3(0 0 0 1) .................................................................74
13.1. Sapphire lattice parameters....................................................................74
13.2. Thermal expansion coefficients of sapphire ...........................................75
13.3. In-plane stress in GaN and GaBN layer: ................................................75
13.4. Discussion: .............................................................................................81

iii
Content
14. Conclusion and Outlook .....................................................................................82
15. Literature ............................................................................................................83
16. Appendix ............................................................................................................87
16.1. Serial port communication ......................................................................87
16.2. Stereographic projection of Silicon (0 0 1)..............................................88
16.3. Stereographic projection of gallium nitride (0 0 1) ..................................89
16.4. Stereographic projection of sapphire (0 0 1)...........................................90
16.5. GaN stress evaluation written in Mathematica: ......................................91

1. Motivation
Motivation
Virtually all types of thin films are expected to contain some amount of residual strain
decisively influencing their mechanical behaviour and, secondary, modifying band-
gaps in semiconductors, transition temperatures in superconductors, magnetic
anisotropy, wear resistance or other important physical parameters. The strains can
be formed unintentionally as an unavoidable consequence of the deposition process
or intentionally in order to control required parameters of devices.
The residual stresses represent very important issue especially in the case of modern
electronics packages representing nowadays complicated composites of
semiconductors, metals, dielectrics and plastics with specific thermal expansion
coefficients, manufacturing temperatures and geometry. For the fabrication as well as
for the practical application of such structures, the control of residual stresses has
turned out to be of utmost importance.
For the production of interconnects in microelectronic chips, aluminium and copper
have been used. When the interconnects are thermally cycled during operation,
various micro-structural effects can occur including grain growth, diffusion, plastic
flow, electromigration etc. All these phenomena are influenced and partly also
controlled by the magnitude of residual stresses in the metals.
On the other hand, residual stresses in semiconductors directly influence important
optical and electronic properties through the deformation of crystal lattice and
subsequently the modification of the band gap parameters. The nitride
semiconductors including the family of refractory materials like indium nitride,
aluminium nitride and especially gallium nitride possess a significant potential for
optoelectronic and piezoelectric applications. The presence of high compressive
residual stresses in nitride-based thin films, however, influence not only the optical
properties but is responsible also for crack formation – a very serious problem in
nitride technology.
The characterization of residual stresses in thin films represent thus a very important
issue for nowadays technology. The main aim of this thesis to perform elevated-
temperature X-ray diffraction characterization of residual stresses in aluminium thin
films and in BGaN/GaN structure focusing micro-structural changes and phenomena
related to intrinsic and extrinsic stresses.
1

2. Introduction
Introduction
Already in ancient times, materials were tested for their reliability. For example the
bending of swords to proof their elasticity and the tapping on ceramic vessels, like
amphora, to detect defects were wide spread testing methods at that time. The first
systematic examinations of materials properties and their quantification have been
reported since the middle age, where this knowledge served for the constructions of
buildings as well as to improve shipbuilding. With the appearance of the first
industrial steam engines in the 19 th century, the demands not only on the material but
also on adequate testing methods raised. Since that time, many important testing
methods have been developed and utilized till nowadays especially for bulk
materials. With the development of microelectronics and with the application of
integrated circuits, new testing methods have been introduced allowing a significant
progress in the miniaturization of the thin film – based structures.
Today’s increasing performance of electronic devices is accompanied by a higher
energy consumption stimulating requirements for the cooling. The heat dissipation
influences not only the electrical, optical and magnetic properties of the devices but
also the magnitude of internal stresses. The presence of residual stresses in thin
films and in sublayers of sandwich structures can not be underestimated due to their
direct influence on all basic physical as well as on mechanical properties of the
devices. Especially in the context of the electromigration in copper or in aluminium
interconnects, the residual stresses represent a very important issue.
Up to now, stresses in thin films have been analyzed predominantly ex situ using
X-ray diffraction, Raman and photoluminescence spectroscopy, the wafer curvature
method, and high resolution transmission electron microscopy. The main practical
advantage of the curvature technique resides in the application for new materials with
unknown elastic constants. The diffraction techniques, on the other hand, are
capable of resolving anisotropic deformation of crystal lattice, thin film behaviour on
anisotropic substrates and stresses even in multilayered structures. Recently, a
significant attention was devoted to the studying of residual stress origins in thin
films. In this case, the elevated-temperature X-ray diffraction provided in important
results leading to the understanding the role of intrinsic-stresses and their formation
in thin films.
Within this thesis structural properties of polycrystalline Al thin film deposited on
Si(1 0 0) substrate and the properties of GaN/BGaN multilayers deposited on c plane
sapphire are studied using elevated-temperature X-ray diffraction technique
implemented recently at Erich Schmid Institute for Materials Science in Leoben. For
the studies of GaN/BGaN/Al2O3(0 0 0 1) structures, a high-resolution monochromator
was used.
2

3. Basic X-ray diffraction expressions
3.1. Reciprocal lattice
Basic X-ray diffraction
A mathematical formalism will be defined to describe scattering phenomena on
crystal structures with translation symmetry [1]. Supposing the crystal periodicity, the
selection of available functions is reduced. A periodical function alone, like sinus or
cosines, is unable to describe the electron density distribution and phase
phenomenon, the basic attributes of X-ray diffraction effect. Fourier series can be
applied to describe the scattering effect.
n
G
= ∫ N(
r)
exp( −i
G . r)
dV
(equ. 3.1)
cell
∑
n ( r ) = nG
exp( i G . r)
(equ. 3.2)
G
The imaginary unit, square root of minus one, is expressed by i. N(r) defines the
electron density within one unit cell while, on the other hand, n(r) denotes the
electron density of the whole crystal. The two vectors r and G in equation 3.1 and 3.2
that appear in the exponential term can be expressed as:
r = x a + y a + z a
(equ. 3.3)
1
1
2
2
3
G = h b + k b + l b
(equ. 3.4)
The summation over G in equation 3.2 should be understood as a summations over
h, k and l from minus infinity to plus infinity. The relation between ai and bi vectors
can be found by calculating the scalar product of G and r. According to the Fourier
series the scalar product must have the following form.
3
G . r = 2π
(h x + k y + l z)
(equ. 3.5)
Considering the equation 3.5, the scalar product of a1 and b2 or a1 and b3 is zero, so
a system of linear equations can be written as:
3

⎛ a1
. b1
⎜
⎜a
2 . b1
⎜
⎝a
3 . b1
The solution of this system is:
a
a
a
1
2
3
b
b
b
. b
. b
2
. b
1
2
3
2
2
a1
. b3
⎞ ⎛1
⎟ ⎜
a2
. b3
⎟ = 2π
⎜0
a ⎟ ⎜
3 . b3
⎠ ⎝0
a2
× a3
= 2π
a . ( a × a )
1
a3
× a1
= 2π
a . ( a × a )
1
a1
× a2
= 2π
a . ( a × a )
1
2
2
2
3
3
3
0
1
0
0⎞
⎟
0⎟
1⎟
⎠
Basic X-ray diffraction
The vectors b1, b2 and b3 of equations 3.7 are called reciprocal lattice vectors.
3.2. Bragg equation
(equ. 3.6)
(equ. 3.7)
The Bragg equation represents a relatively simple way to describe the scattering
phenomenon on a crystal lattice. Consider an incident beam that is reflected by a
family of net planes. It’s worth to mention that the path difference between
neighbouring net planes causes a phase difference between the diffracted beams, so
the reflected radiation shows constructive and destructive interference. The Bragg
equation, formula 3.8, predicts that constructive interference only occurs if the ratio
between the phase difference and wavelength is an integer value of n.
2 d sin( θ)
n =
(equ. 3.8)
λ
The variable θ is the angle between the incident or the reflected beam and the net
plane. Because elastic scattering is assumed, the incident and reflected beam must
have same wavelength λ. The last parameter is the lattice spacing d, the shortest
distance between two neighbouring net planes with same Miller indices (h k l) and the
surface normal G that is inversely proportional to the plane spacing d.
π
=
G
2
d (equ. 3.9)
4

Basic X-ray diffraction
Equation 3.9 is valid for all crystal systems, and the components of G can be
replaced by the points of intersection u1, u2 and u3 of the net plane with the elongated
translation vectors a1, a2 and a3.
Figure 3.1: Crystallographic plane and surface normal
The components of G are for that reason:
1
h =
u1
1
k =
u2
1
l = (equ. 3.10)
u
3
5

4. Mechanical properties of crystals
4.1. Stress, strain and displacement
Mechanical properties of crystals
Internal stresses are produced by external forces acting on a body [2]. These
external forces can be separated into two categories, like the distribution of forces
over the surface, such as hydrostatic pressure, and distributed forces over the
volume, for example gravitational forces or magnetic forces. A motion of the body is
also able to influence the internal stresses. The most important expression to
describe stresses can be derived by assuming the conservation of impulse.
ρ a = div(
σ)
+ f
(equ. 4.1)
Where ρ is the density of the body, a is the acceleration, f denotes the distributed
forces over the volume and σ, a second ranked tensor, represents the internal
stresses.
σ
σ n
n = (equ. 4.2)
The distributed forces or stresses over a surface are written in equation 4.2 that is the
product between the stress tensor and the surface normal vector n, and can be
understood as a boundary condition for equation 4.1. In the case of stress
measurement on thin films, which is a static problem, the acceleration is zero and the
volume forces are neglected.
div ( σ ) = 0
(equ. 4.3)
A body under stresses will deform, this means that a point P of the unstressed body
moves to the position P’. The vector connecting P and P’ is called displacement u.
We consider a short and a long rod with same cross section and made of the same
material. Let us say that the change of length is the displacement, than one can see
that the displacement of the longer rod is greater than the displacement of the small
rod, if the same stress is applied at the ends of both rods. An assessment based on
strains to characterize the deformation will show same results for both rods. This
means that the influence of geometry does not play a role.
6

ε
ij
=
1 ⎛
⎜
∂ ui
2 ⎜
⎝ ∂x
j
∂ u j ⎞
+ ⎟
∂x
⎟
i ⎠
Mechanical properties of crystals
(equ. 4.4)
Equation 4.4 defines the strain tensor for small displacements. A closer look on the
strain tensor reveals the symmetric property. It was not mentioned above but the
stress tensor is also symmetric.
4.2. Anisotropic Elasticity
ε = ε
(equ. 4.5)
ij
ij
ji
σ = σ (equ. 4.6)
Further equations are needed to solve a general mechanical problem, because if all
unknown quantities of the previous chapter are count together the total sum will be
15, in detail there are six stresses, just as much strains and additionally three
displacements. On the other side we have three expressions to describe stresses
(equation 4.1 or 4.3) and six formulas of strain/displacement relations (equation 4.4),
so remains a lack of six equations.
The missing equations are based on the mechanical behaviour of a material that is
assumed to be elastic. In literature a material is often treated in an isotropic way. It
means that a property like the electrical resistant or thermal expansion coefficient is
not depending on the materials direction. However, a crystal is anisotropic [3], and
therefore a general expression will define the elastic stresses/strains relations.
ij
ijkl
ji
σ = c ε
(equ. 4.7)
ij
ijkl
kl
ε = s σ
(equ. 4.8)
The stiffness tensor cijkl of equation 4.7 is the relation between stress and strain
tensor, both of second rank including nine components, thus the stiffness tensor must
consist of 81 elements. Very similar to the stiffness is the compliance tensor sijkl in
equation 4.8. It is simple to find a relation between both expressions.
−1
c = s
kl
−1
s = c
(equ. 4.9)
7

Mechanical properties of crystals
Because of the anisotropic material behaviour, with unequal properties in different
directions, the position of the crystal system with reference to the sample is of
interest. Generally the crystal and sample system do not coincide due to angular
differences. In order to express the anisotropic behaviour in a sample system, the
compliance or stiffness tensors must be rotated.
A ' = O O A
(equ. 4.10)
For a fourth ranked tensor the expression has the following form:
ijkl
ij
im
ik
jn
jl
ko
kl
A ' = O O O O A
(equ. 4.11)
In literature a second preferred way to express the relation between stresses and
strains according to W. Voigt [4] is often used. By applying the matrix notation, the
stress and strain tensors are reduced to vectors with six components, thus the
compliance and stiffness matrices have 36 components. A comparison between
stresses and strains written in matrix notation and tensor notation shows:
⎛ ε
⎜
⎜ε
⎜
⎝ ε
⎛ σ
⎜
⎜σ
⎜
⎝ σ
11
12
13
11
12
13
ε
ε
ε
12
22
23
σ
σ
σ
12
22
23
ε
ε
ε
13
23
33
σ
σ
σ
13
23
33
⎞ ⎛ σ
⎟ ⎜
⎟ = ⎜σ
⎟ ⎜
⎠ ⎝ σ
⎛
⎜ ε1
⎞
⎟
⎜
⎜ 1
⎟ = ε
⎟
⎜ 2
⎠ ⎜ 1
⎜ ε
⎝ 2
6
5
1
6
5
lp
1
2
σ
σ
σ
ε
1
2
ε
2
ε
6
2
4
mnop
6
4
σ5
⎞
⎟
σ4
⎟
σ ⎟
3 ⎠
1
ε
2
1
ε
2
ε
3
5
4
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(equ. 4.12)
(equ. 4.13)
The first two and last two suffixes of the tensor notation are merged to a single
number running from one to six, according to the scheme:
Tensor
notation
Matrix
notation
11 22 33 23, 32 31, 13 12, 21
1 2 3 4 5 6
8

The relations between the compliance tensor and matrix are:
sijkl = smn when m and n are 1, 2 or 3.
2 sijkl = smn when either m or n are 4, 5 or 6.
4 sijkl = smn when both m and n are 4, 5 or 6.
Mechanical properties of crystals
The advantage of working with matrix notation is the compacter formalism, though
the rotation of the compliance and the stiffness matrix is more complicated than the
rotation in the tensor notation. For example an isotropic material can be
characterized by the following stiffness matrix:
⎛ c11
c12
c12
0 0 0 ⎞
⎜
⎟
⎜c12
c11
c12
0 0 0 ⎟
⎜c
⎟
12 c12
c11
0 0 0
c = ⎜
⎟
(equ. 4.14)
⎜ 0 0 0 c44
0 0 ⎟
⎜ 0 0 0 0 c ⎟
44 0
⎜
⎟
⎜
⎟
⎝ 0 0 0 0 0 c44
⎠
Isotropic materials have only two independent components, so the third one can be
calculated:
4.3. Intrinsic and extrinsic stress
1
c44 = ( c11
− c12
)
2
The residual stresses in thin films results from a growth procedure and from a cooling
down to the operation temperature after the deposition with specific intrinsic and
extrinsic stress contributions, respectively. The intrinsic stresses originate from a
specific microstructure development and the film densification during the growth [5].
An atomic disorder caused by foreign atoms integrated into the lattice can serve as
an example of the intrinsic stress origin. An impurity atom can substitute a lattice
atom or it can be found in lattice gaps as an interstitial atom. The compressive/tensile
stress is getting higher if the atomic radius of the impurity atoms is larger/smaller than
the atomic radius of the original lattice atoms. Intrinsic stresses can be reduced by
applying high substrate temperatures during deposition, because of the high mobility
of the atoms reduces the disorder. An additional part of intrinsic stresses occur in
sputtered layers. The incoming sputtered atoms are hitting the layer and this is like
9

Mechanical properties of crystals
shot peening, setting the layer under compressive stress. This effect is known as
atomic peening.
Extrinsic stresses originate from the cooling down procedure and basically depend on
the different constants of thermal expansion of layer and substrate. After cooling or
heating, starting from the deposition temperature TS, the layer material is elastically
or plastically deformed. In the elastic region the extrinsic stress is:
hkl
σ = ( α − α ) ( T − T )
(equ. 4.15)
ex
M l s
S
The thermal expansion coefficient αl belongs to the layer, the other one αs to the
substrate and the quantity M hkl is the biaxial modulus of the layer [6] that is
depending on the crystallographic orientation. The final stresses are the sum of
intrinsic and extrinsic stresses.
Residual stresses are reduced through crack formation under tensile stress,
delamination of the layer or plastic deformation forming hills under compressive
stress.
10

5. Orthogonal tensors and rotation
5.1. Basic relations
Orthogonal tensors and rotation
Crystals are anisotropic bodies, which means that the mechanical behaviour depends
on the crystal orientation. For this reason, it is important to exactly define the
orientation of the crystal in the sample coordinate system. Generally the sample and
the crystal coordinate systems can be related by a rotation (Chapter 4.2) expressed
by an orthogonal tensor O.
We consider two right angled coordinate systems with the unit vectors ei and ei’
(figure 5.1). A unit vector belonging to one coordinate system, for example ei, is
projected on the axes of the other system so that the position of this vector can be
obtained in the new coordinate system ei’ [7]. The length of the projected unit vector
on one axis is:
Figure 5.1: Relation between two tilted coordinate systems
e i ' . e j = ei
' e j cos( ei
',
e j)
e i ' = 1
j 1 = e
e . e = cos( e ',
e )
(equ. 5.1)
i ' j
i j
By applying equation 5.1 on all unit vectors of one coordinate system, an orthogonal
tensor is received that contains only cosine functions. Such a expression is also
known as direction cosine. For example let us use e1 to express e1‘, where the
components of e1‘ and e1 are x1’, y1’, z1’ and x1, y1, z1.
11

⎛x
1'⎞
⎛ cos( e1'
, e1)
⎜ ⎟ ⎜
⎜ y1'⎟
= ⎜cos(
e 2'
, e1)
⎜ ⎟ ⎜
⎝ z1'
⎠ ⎝cos(
e3'
, e1)
5.2. Rotation in a plane
cos( e ',
e )
1
2
3
2
cos( e ',
e )
2
cos( e ',
e )
1
1
2
Orthogonal tensors and rotation
cos( e1'
, e3
) ⎞ ⎛ x1
⎞
⎟ ⎜ ⎟
cos( e 2'
, e3
) ⎟ ⎜ y1
⎟
cos( e ⎟ ⎜ ⎟
3'
, e3
) ⎠ ⎝ z1
⎠
(equ. 5.2)
e ' = O e
(equ. 5.3)
To learn more about how a rotation around a certain axis will affect the position of the
sample, it is necessary to consider a body on which points the vector r, like in figure
5.2. A positive rotation of the body is synonymous with the rotation of the vector r
around the e3 axis and at the end of the operation the vector r coincides with r’.
The new point r’ with the components x’, y’ and z’ can be expressed by the starting
point r:
r ' = O
3
r
⎛cos(
ϕ)
− sin( ϕ)
0⎞
⎜
⎟
3
O = ⎜ sin( ϕ)
cos( ϕ)
0⎟
(equ. 5.4)
⎜
⎟
⎝ 0 0 1⎠
The orthogonal tensors, that are describing a rotation around the two other axes, are
shown in equation 5.5 and 5.6.
Figure 5.2: Rotation of an object
⎛1
0 0 ⎞
⎜
⎟
1
O = ⎜0
cos( ψ)
− sin( ψ)
⎟
(equ. 5.5)
⎜
⎟
⎝0
sin( ψ)
cos( ψ)
⎠
12

Orthogonal tensors and rotation
⎛ cos( ω)
0 sin( ω)
⎞
⎜
⎟
2
O = ⎜ 0 1 0 ⎟
(equ. 5.6)
⎜
⎟
⎝−
sin( ω)
0 cos( ω)
⎠
The body in position r’ can be turned back by a rotation around e3 axis in the
negative direction. Thus the angle’s sign is negative and the body’s starting position
is supposed to be r’.
⎛x
⎞ ⎛ cos( ϕ)
⎜ ⎟ ⎜
⎜ y⎟
= ⎜ − sin( ϕ)
⎜ ⎟ ⎜
⎝ z ⎠ ⎝ 0
sin( ϕ)
cos( ϕ)
0
0⎞
⎛ x'⎞
⎟ ⎜ ⎟
0⎟
⎜ y'⎟
1⎟
⎜ ⎟
⎠ ⎝ z'
⎠
A closer look shows that the orthogonal tensor is the transposed or the inverse
version of O 3 .
5.3. Euler angles
3 T
( O ) r'
r = (equ. 5.7)
In the previous chapter we only thought about a rotation in a plane. In practice more
possibilities are needed to express the tilt between two coordinate systems. One
possibility is the concept of Euler angles [8], that can be understood as three
separated rotations around certain axes, similar like before. The first step is the
rotation around the e3 axis so e1 and e2 will change their position. After that the new
e1 is supposed to be the next rotation axis, which will lead to two new e2 and e3 axes,
and the last step is like the first one.
After the first rotation the new axis is e1.
⎛cos(
ϕ1)
− sin( ϕ1)
0⎞
⎜
⎟
ei
' = ⎜ sin( ϕ1)
cos( ϕ1)
0⎟
e
⎜ 0 0 1⎟
⎝
⎠
⎛1
0 0 ⎞
⎜
⎟
ei
'' =
⎜0
cos( φ)
sin( φ)
⎟ ei
'
⎜0
sin( ) cos( ) ⎟
⎝ − φ φ ⎠
i
13

The last step is the rotation around the new e3 axis.
All equations written in full are:
⎛cos(
ϕ2
) − sin( ϕ2
) 0⎞
⎜
⎟
ei
'' ' = ⎜ sin( ϕ2
) cos( ϕ2
) 0⎟
ei
''
⎜ 0 0 1⎟
⎝
⎠
Orthogonal tensors and rotation
⎛cos(
ϕ2
) − sin( ϕ2
) 0⎞
⎛1
0 0 ⎞ ⎛cos(
ϕ1)
− sin( ϕ1)
0⎞
⎜
⎟ ⎜
⎟ ⎜
⎟
ei
' ' ' = ⎜ sin( ϕ2
) cos( ϕ2
) 0⎟
⎜0
cos( φ)
sin( φ)
⎟ ⎜ sin( ϕ1)
cos( ϕ1)
0⎟
e
⎜ 0 0 1⎟
⎜0
sin( ) cos( ) ⎟ ⎜ 0 0 1⎟
⎝
⎠ ⎝ − φ φ ⎠ ⎝
⎠
After the multiplications the orthogonal Euler tensor has the following formula:
e
O
⎛ cos( ϕ1)
cos( ϕ2
) − cos( φ)
sin( ϕ1)
sin( ϕ2
)
⎜
= ⎜−
cos( ϕ1)
sin( ϕ2
) − cos( φ)
sin( ϕ1)
cos( ϕ2
)
⎜
⎝
sin( φ)
sin( ϕ1)
sin( ϕ ) cos( ϕ ) + cos( φ)
cos( ϕ ) sin( ϕ )
1
− sin( ϕ ) sin( ϕ ) + cos( φ)
cos( ϕ ) cos( ϕ )
1
2
2
− sin( φ)
cos( ϕ )
1
1
1
2
2
sin( φ)
sin( ϕ2
) ⎞
⎟
sin( φ)
cos( ϕ2
) ⎟
cos( φ)
⎟
⎠
i
(equ. 5.8)
14

Basic Physical Properties
6. Basic Physical Properties of layer and substrate materials
6.1. Polycrystalline aluminium on monocrystalline silicon
6.1.1. Properties of aluminium layer
Aluminium ore, most commonly bauxite, occurs mainly in tropical and sub-tropical
areas. The raw material bauxite is converted into alumina in the Bayer process, which
is reduced to aluminium metal in electrolytic cells known as pots by adding cryolite [9].
Pure aluminium shows no phase transformation in the solid state and will crystallise at
an equilibrium temperature of about 660°C in a closed packed face centred lattice [10]
with a mono-atomar basis (figure 6.1).
As basis for the stress evaluation the anisotropic elastic behaviour of a cubic material
is represented in equation 6.1, where the low number of independent elastic constants
corresponds to the high symmetry of the cubic system [3, 4].
⎛s11
⎜
⎜s12
⎜s12
s = ⎜
⎜ 0
⎜ 0
⎜
⎝ 0
s
s
s
12
11
12
0
0
0
s
s
s
12
12
11
0
0
0
s
0
0
0
44
0
0
Figure 6.1: Cubic face centred unit cell
s
0
0
0
0
44
0
0 ⎞
⎟
0 ⎟
0 ⎟
⎟
0 ⎟
0 ⎟
⎟
s ⎟
44 ⎠
⎛ c
⎜
⎜c
⎜c
c = ⎜
⎜ 0
⎜ 0
⎜
⎝ 0
11
12
12
c
c
c
12
11
12
0
0
0
c
c
c
12
12
11
0
0
0
c
0
0
0
44
0
0
c
0
0
0
0
44
0
0 ⎞
⎟
0 ⎟
0 ⎟
⎟
0 ⎟
0 ⎟
⎟
c ⎟
44 ⎠
(equ 6.1)
15

Basic Physical Properties
The components of the compliance and stiffness matrix or tensor are not real constant
values due to their temperature dependence. In figure 6.2 the three independent
compliance matrix components of aluminium were plotted [11]. Another characteristic
parameter associated with the temperature influence is the coefficient of thermal
expansion that has a value of 23,8 10 -6 K -1 for aluminium [12].
s 12 / [10 -3 GPa -1 ]
-5
-6
-7
-8
-9
-10
s 11 / [10 -3 GPa -1 ]
24
22
20
18
16
14
34
0 100 200 300 400 500
6.1.2. Properties of silicon substrate
The Czochralski technique is a widespread production method capable of providing
silicon single crystals with a relatively large size. After cutting the silicon single crystals
into thin wafers, followed by surface treatments like lapping, etching and polishing,
they are ready to serve as substrate for thin film deposition.
Silicon has a diamond-like structure [1, 10], which is based on a face centred cubic
lattice with the primitive basis consisting of two atoms at position [0 0 0] and [¼¼¼].
This basis reduces therefore the symmetry but it is not changing the general elastic
mechanical behaviour [13] of the cubic lattice (figure 6.3) described by the compliance
or stiffness matrix (at room-temperature).
T / °C
Figure 6.2: Temperature dependent compliance constants of aluminium
c11 = 165,64 GPa ; c12 = 63,94 GPa ; c44 = 79,51 GPa;
s 11
s 12
s 44
50
48
46
44
42
40
38
36
s 44 / [10 -3 GPa -1 ]
16

Basic Physical Properties
During cooling, starting from the melting point at 1410°C, silicon crystallises in the
diamond structure without showing allotropic transformation in the solid state. The
cooling process is associated with the thermal contraction of the silicon material that
has a thermal expansion coefficient of 2,616 10 -6 K -1 at room temperature [13].
6.2. Monocrystalline GaN on monocrystalline sapphire
6.2.1. GaN
Unfortunately, bulk crystals of nitrides cannot be obtained by conventional methods of
liquid phase epitaxy, because of extremely high melting temperatures and very high
decomposition pressures at the melting point.
Several production techniques are available for growing thin gallium nitride (GaN)
films. These methods can be divided in two groups, one where chemical reactions play
an important role and the other one where only physical process are responsible for
the layer formation. For example a representative of the first group would be
metalorganic vapour deposition (MOCVD) technique or reactive sputtering and for
instance a production method belonging to the second group would be molecular
beam epitaxy (MBE).
Figure 6.3: Diamond structure
17

Basic Physical Properties
Group III nitrides like AlN, GaN and InN can crystallize in wurtzite, zinc-blende and
rock-salt crystal structures. At ambient conditions the thermodynamically stable phase
is the wurtzite structure and only at higher pressures an allotropic transformation
changes the crystal to rock salt structure. The zinc-blende structure is metastable and
may be stabilized by heteroepitaxial growth on substrates reflecting topological
compatibility.
In the preparation phase of the experiment several Bragg reflections of the thin layer
were measured, leading to the result that the thin GaN film is based on wurtzite
structure (figure 6.4), where the anions form a closed packed hexagonal structure and
the cations with smaller atomic radius will occupy the tetrahedral gaps [10]. Owing to
the stoichiometry only the half of all available tetrahedral positions can be filled.
GaN crystallized in wurtzite structure and therefore it has a lower symmetry than
native hexagonal structures. All hexagonal based lattices have same compliance and
stiffness matrix [3] that are expressed in equation 6.2.
⎛s11
⎜
⎜s12
⎜s13
s = ⎜
⎜ 0
⎜ 0
⎜
⎝ 0
s
s
s
12
11
13
0
0
0
s
s
s
13
13
33
0
0
0
s
0
0
0
44
0
0
s
0
0
0
0
44
0
2 ( s
Figure 6.4: Wurtzite structure
11
0
0
0
0
0
− s
11
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
) ⎟
⎠
⎛ c11
⎜
⎜c12
⎜c13
⎜
c =
⎜ 0
⎜
0
⎜
⎜ 0
⎝
c
c
c
12
11
13
0
0
0
c
c
c
13
13
33
0
0
0
c
0
0
0
44
0
0
c
0
0
0
0
44
0
1
( c
2
11
0
0
0
0
0
− c
11
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
) ⎟
⎠
(equ 6.2)
18

Basic Physical Properties
During the measurement the temperature is increased or decreased, and it must be
taken into account that the elastic stiffness tensor has no longer constant values. In
the evaluation of the data, the temperature dependence of the elastic constants,
measured by R. R. Reeber and K. Wang [14], will be taken into consideration.
c 13 / GPa
100
98
96
94
92
c11 / GPa
c33 / GPa
390
385
380
375
370
365
360
134
200 300 400 500 600 700 800 900 1000
T / °C
Figure 6.5: Temperature dependent stiffness constants of wurtzite GaN
c 11
c 33
c 12
c 13
c 44
146
144
142
140
138
136
c 12 / GPa
99,0
98,5
98,0
97,5
97,0
96,5
c 44 / GPa
19

Basic Physical Properties
The plane spacing of hexagonal lattices is expressed through h, k, l, a0 and c0. Both
unknown unstressed lattice parameters a0 and c0 must be considered when refining
the structural parameters. R.R. Reeber and K. Wang [15] examined the lattice
parameters of 99,99% pure and annealed GaN powder using neutron scattering. The
results of their experiment can be seen in figure 6.6. With this data a relation between
a0 and c0 is found by expressing c0 through the lattice parameter a0 and the ratio c0/ a0
given by the neutron scattering measurement.
-1
c 0 a 0
1,6263
1,6262
1,6261
1,6260
1,6259
1,6258
1,6257
1,6256
a 0 / A
3,200
3,198
3,196
3,194
3,192
3,190
3,188
3,186
200 300 400 500 600 700 800 900 1000
T / K
a 0 spacing
c 0 spacing
-1
c0 a0 Figure 6.6: Lattice parameters of wurtzite GaN
5,195
5,190
5,185
5,180
c 0 / A
20

6.2.2. Boron nitride
Basic Physical Properties
Boron nitride (BN) has different properties than other group III nitride members [16].
The crystal structures and related physical properties are analogous to modifications
of carbon, thus boron nitride exists in graphite like hexagonal structure. The former
includes the stable zinc-blende and metastable wurtzite crystal structure. In contrast to
the other group III nitrides no transition to the rock-salt structure at high pressures has
been observed. The hexagonal boron nitride has outstanding mechanical properties,
but it is less interesting for electronic applications.
Figure 6.7: Modifications of boron nitride [17]
gBN … hexagonal BN, zBN … zinc-blende BN, wBN … wurtzite BN
The buffer layer contains only a small fraction of boron nitride, thus it is supposed that
the boron will substitute gallium, so the buffer layer will also crystallise in wurtzite
structure. The different lattice constants between boron nitride (a0=2,553A and
c0=4,228A) and gallium nitride (a0=3,188A and c0=5,185A) [16] will cause tension in
the buffer layer because of the smaller atomic radius of boron.
21

Basic Physical Properties
The elastic stiffness constants for wurtzite boron nitride at 300K, according to K. Kim,
W.R.L. Lambrecht and B. Segall [18], are:
c11 = 987 GPa; c12 = 143 GPa; c13= 70 GPa; c33 = 1020GPa; c44= 369GPa;
6.2.3. Sapphire substrate
Sapphire is the most used substrate for the growth of group III nitrides, which can be
produced by the Czrochalski technique with high crystal quality and at low cost.
Based on the 2:3 stoichiometry aluminium cations that take an octahedral position
must fill two third of available sites [19]. To see how this occur a cation layer between
two layers of close packed oxygen ions is drawn in figure 6.8.
Figure 6.8: Shifting of aluminium cation layer
These octahedral sites occupied by aluminium ions form a hexagonal array with the
same spacing as the oxygen layer. The next cation layer has the same honeycomb
configuration but is shifted by one atomic spacing in the direction of the vector 1. After
another close packed oxygen layer, a third cation layer is placed, that is shifted by the
vector 2. If a vertical slice as indicated by the dashed line is taken than the
arrangement of cathions is drawn like in figure 6.9. The columns of octahedral sited
perpendicular to the (0 0 0 1) plane alternate in having every two sides occupied and
one empty. Only by considering the stacking of the closed packed oxygen ion layers
must follow that the sapphire crystal is based on a hexagonal lattice. The anisotropic
22

Basic Physical Properties
elastic behaviour is for that reason expressed by equation 6.2. The following stiffness
constants for sapphire are valid for room temperature [20].
c11 = 496 GPa; c12 = 164 GPa; c13= 115 GPa; c33 = 498GPa; c44= 148GPa;
A thermal property influencing the origin of extrinsic stress is the thermal expansion
coefficient that is distinguished between expansion parallel and perpendicular to
c-axis. According to Landolt and Börnstein [21] the thermal expansion coefficients for
sapphire are 7,5 10 -6 K -1 perpendicular to c-axis and 8,5 10 -6 K -1 parallel to c-axis.
Figure 6.9: (1 0 1 0) vertical slice according to dashed line in figure 6.8
23

7. Polycrystalline and monocrystalline Model
7.1. Calculating isotropic elastic constants
7.1.1. Isotropic material
Polycrystalline and monocrystalline Model
An untextured material will behave in an isotropic way due to the randomly oriented
grains, though a grain can be seen as single crystal with anisotropic elastic
properties. For calculating stresses in isotropic materials, the Hill model [22] can be
used which is based on Voigt and Reuss approach.
7.1.2. Voigt model
W. Voigt [4] assumed an untextured polycrystalline material where all grains are
under the same strain. Starting with the matrix notation of the stiffness matrix for a
cubic material (equation 6.1) and converting the expression into tensor notation
(chapter 4.2) a rotation can be performed using Euler angles (equation 5.8) to rotate
the crystal in any possible position. A mean value of all random orientated crystals is
taken to express an isotropic behaviour (equation 4.14) of an untextured cubic
material.
2π
π 2π
V 1
e e e e
c ijkl = ∫ ∫ ∫ Oim
O jn Oko
Olp
cmnop
sin( φ)
dϕ1
dφ
dϕ2
(equ 7.1)
8π
0 0
c
c
c
0
V
1111
V
1122
V
2323
= c
= c
= c
V
11
V
12
V
44
1
= ( 3c
5
1
= ( c11
5
1
= ( c11
5
11
+ 2 c
+ 4 c
− c
12
12
12
+
−
+
3 c
4 c
2 c
44
44
)
44
)
)
(equ 7.2)
The stiffness form is unusable for calculating strains, thus the elastic constants must
be converted into compliance form by applying equation 4.9. First the stiffness
components of the cubic system must be replaced on the right side, and afterwards
the isotropic stiffness constants of the Voigt average must be expressed by
compliance components of an isotropic material, that has only two independent
parameters.
s
V
44
=
2 ( s
V
11
− s
V
12
)
24

Polycrystalline and monocrystalline Model
Therefore the compliance components of the Voigt average defined by compliance
constants of a cubic material are:
s
V
12
s
V
11
=
= 2 s
11
1 ⎛
⎜
⎜−
s
2 ⎝
s
11
− s
12
+ 3s
2
5 ( s11
− s12
)
−
3s
− 3s
+ s
12
5 ( s
11
12
44
2
5 ( s11
− s12
)
+
3s
− 3s
+ s
− s
11
) s
12
44
⎟ ⎞
⎠
V
11 12 44
44 = (equ 7.3)
3s11
− 3s12
+ s44
The sample can be rotated around three different angles ω, ψ and ϕ, but the rotation
cannot be performed in any order of these angles. In figure 7.1 the sample and two
coordinate systems are shown. The first one is the laboratory coordinate system eL
which will be fixed and independent of the sample rotation. In this system the plane
spacing is measured, because G will always be parallel to eL3. The second one is the
sample coordinate system eS that is associated with the sample and will change
position during rotation. The problem is, that the strains are measured in the
laboratory system, but the stresses must be expressed in the sample system, so a
rotation order must be performed to describe the sample’s tilt during the
measurement. Let us start at a position where both systems coincide. The first
rotation performed by the diffractometer is around eS1 direction by an angle ω,
marked with the double arrow. In this position the new rotation is done around the
new eS2 axis and the last one around the new eS3 axis. This is similar to the Euler
angles procedure.
a.) starting position b.) ψ-rotation c.) ϕ-rotation d.) end position
and ω-rotation
Figure 7.1: Rotation of the sample with respect to the laboratory system
According to chapter 4.2 the laboratory system can be transformed by rotation into
the sample system.
25

e =
The transformation matrix written in full is:
Polycrystalline and monocrystalline Model
3 1 2
D
S = O O O e L O e L
(equ 7.4)
⎛cos(
ϕ)
cos( ω)
− sin( ϕ)
sin( ψ)
sin( ω)
− sin( ϕ)
cos( ψ)
cos( ϕ)
sin( ω)
+ sin( ϕ)
sin( ψ)
cos( ω)
⎞
⎜
⎟
D
O = ⎜sin(
ϕ)
cos( ω)
+ cos( ϕ)
sin( ψ)
sin( ω)
cos( ϕ)
cos( ψ)
sin( ϕ)
sin( ω)
− cos( ϕ)
sin( ψ)
cos( ω)
(equ 7.5)
⎟
⎜
⎟
⎝ − cos( ψ)
sin( ω)
sin( ψ)
cos( ψ)
cos( ω)
⎠
Equation 7.4 will transform the laboratory system into the sample system, but for the
evaluation of stresses the opposite way is demanded. For that reason the inverse
orthogonal tensor O D is used.
D D
σ = O O σ
ϕψ (equ 7.6)
ij
ki
The change of the orthogonal tensor’s suffix, in equation 7.6, represents the
transposed or inverted version of O D . The thin film is considered to be under plane
stress, which means that every component of the stress tensor on the right side that
has at least one three as suffix is zero.
ε
ϕψ,
V
33
=
lj
kl
V ϕψ
s33kl σkl
In the experiment the ω−angle has no significance and it is set to zero. After
simplifying the expression for the Voigt model is:
= ( σ
) V
ϕψ,
V
ε33 11 + σ22
1 + σϕ
2 ( s
V =
1
σ ϕ
11
− s
V
2
12
V
) ( s11
+ 2 s12
) − ( s
2 (3s
− 3s
+ s
11
12
2
11
44
5 ( s11
− s12
) s44
=
2 (3s
− 3s
+ s
11
12
44
sin
2
− 3s
)
)
( ψ)
2
2
= σ11
sin ( ϕ)
− σ12
sin( 2 ϕ)
+ σ22
cos ( ϕ)
(equ 7.7)
12
) s
44
26

7.1.3. Reuss model:
Polycrystalline and monocrystalline Model
In contrast to Voigt supposes Reuss that all grains of an untextured material are
under same stresses, so only grains with a net plane normal parallel to eL3 will diffract
[23]. Therefore the crystal must be rotated in a position where the net plane is
perpendicular to the measuring direction eL3.
v
3
h b1
+ k b2
+ l b
=
h b + k b + l b
v
2
1
l a
=
l a
1
2
2
2
− k a
− k a
v = v × v
The wanted net plane normal can be rotated into a parallel position to eL3 by using
the direction cosine.
⎛ ec1
. v1
ec2
. v1
ec3
. v1
⎞
⎜
⎟
e L = ⎜e
c1 . v 2 ec2
. v2
ec3
. v2
⎟ eC
= O
⎜e
c1 . v3
ec2
. v3
ec3
. v ⎟
⎝
3 ⎠
The following othogonal tensor O hkl is only valid for cubic systems.
O
hkl
⎛
⎜
⎜
⎜
= ⎜
⎜
⎜
⎜
⎝
h
h
2
2
k
2
+ k
+ l
0
h
2
+ k
2
2
+ l
2
+ l
2
−
k
2
+ l
h
+ k
2
2
k + l
k
2
2
h k
h
l
2
2
+ k
2
+ l
2
3
2
3
3
+ l
2
3
3
−
k
hkl
2
e
−
C
+ l
h
2
2
h l
h
k
+ k
2
2
k + l
l
An untextured material will have lots of grains with G parallel to eL3 so in every
position around the net plane normal the same amount of crystals is present.
O
λ
⎛cos(
λ)
⎜
= ⎜ sin( λ)
⎜
⎝ 0
− sin( λ)
cos( λ)
0
0⎞
⎟
0⎟
1⎟
⎠
2
+ k
2
+ l
2
2
+ l
2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
27

2π
hkl,
R 1 λ λ λ λ λ λ λ λ
sijkl =
2π
∫ Oim
O jn Oko
Olp
Omq
Onr
Oos
Opt
sqrst
0
Polycrystalline and monocrystalline Model
After calculating the Reuss average the mean elastic constants are dependent of the
miller indices h, k and l. The strain is written in the following way, by using the
stresses in the laboratory system from before:
After simplifying:
ϕψ,
hkl,
R hkl,
R ϕψ
ε33 = s33 kl σkl
ϕψ , hkl,
R
hkl
ε33 = ( σ11
+ σ22
) R1
+ σϕ
R
R
7.1.4. Hill model
R
hkl
2
hkl
1
= s
= s
11
12
− s
+ Γ
12
⎛
⎜s
⎝
11
− s
⎛
+ 3 Γ ⎜s
⎝
11
12
hkl
2
s
−
2
− s
12
44
2
sin ( ψ)
⎞
⎟
⎠
s
−
2
44
⎞
⎟
⎠
dλ
2 2 2 2 2 2
h k + k l + h l
Γ =
(equ 7.8)
2 2 2 2
( h + k + l )
The Reuss and Voigt model are both limits of elastic stresses. In the hill model, that
shows good correspondence with practical results, simply the average value of both
limits is taken by allocating x in equation 7.9 to be a half.
ε = x ε + ( 1 − x)
ε
(equ 7.9)
hill
33
ϕψ,
hkl,
R
33
The strain in direction of the net plane normal is:
After rewriting equation 7.10 follows:
ε
hill
33
ϕψ
d − d
=
d
0
0
ϕψ,
V
33
(equ 7.10)
28

d
ϕψ
ϕψ,
hkl,
R
= d (1+
( x ε + ( 1−
x)
ε
0
33
Polycrystalline and monocrystalline Model
ϕψ,
V
33
Equation 7.11 will be used for nonlinear regression and d0 is substituted by:
d
ϕψ
a 0 ϕψ,
hkl,
R
ϕψ,
V
= (1+
( x ε33
+ ( 1−
x)
ε33
)) (equ 7.11)
2 2 2
h + k + l
7.2. Calculating stresses for a single crystalline material
The examination of isotropic polycrystalline material differs from anisotropic single
crystalline material in such a way that an untextured polycrystalline material fulfils the
Bragg condition at every ψ-ϕ tilt because there are always crystals with net plane
normals parallel to the measuring direction in the laboratory system. However Bragg
reflections of single crystals can only be detected at strict defined tilts [24].
ε = O O O O s σ
(equ 7.12)
ij
e
mi
e
nj
e
ok
Only strains can be measured, hence the stresses must be expressed with a linear
elastic anisotropic material model. The strains in the sample system are depending
on the crystal position therefore the orthogonal tensor O e , which is described by Euler
angles, is introduced to express the relation between the sample and the crystal
system.
ij
D
ki
e
pl
D
lj
kl
mnop
kl
))
ε = O O ε
ϕω (equ 7.13)
The strain measurement is done in the laboratory system, so the sample’s tilt is
defined by equation 7.13 which is similar to equation 7.6 except that the strain tensor
is rotated and ω−angle is used instead of ψ.
In the case of gallium nitride and gallium boron nitride on sapphire the basal plane of
the GaN crystal and BGaN crystal is perpendicular to the surface normal, so further
calculations are simplified in such a way that no orthogonal tensor O e is needed,
because of the transversal isotropy of the hexagonal lattice.
ε = O O s σ
ϕω (equ 7.14)
33
D
i3
D
j3
ijkl
kl
29

Polycrystalline and monocrystalline Model
If we are assuming a biaxial stress state than all stresses that have at least one three
as suffix will be zero.
ε ϕ
= ( s
11
σ
11
+ s
12
σ
22
2
) cos ( ϕ)
+ ( s
12
ε
= ε
+ ε
sin ( ω)
ϕω
33
C ϕ 2
ε
σ
c
11
= s13 ( σ11
+ σ22
)
+ s
11
σ
22
2
) sin ( ϕ)
+ ( s
11
− s
12
) σ
12
sin(2 ϕ)
− s13
( σ11
+ σ
(equ 7.15)
The plane spacing is calculated by applying the Bragg equation and can be
expressed by the two lattice parameters a and c of the hexagonal lattice.
The strain parallel to G is:
2
2 2
1 4 (h + h k + k ) l
= +
(equ 7.16)
2
2
2
d 3a
c
ε
ϕω
33
ϕω
d − d
=
d
The final regression formula has the following form:
d
0
0
ϕω
C ϕ
= d0
(1+
ε ) = d0
(1+
ε + ε sin ( ω)
)
(equ 7.17)
ϕω 2
33
It is possible to replace one lattice parameter by using the ratio of cN0 and aN0
measured by R. R. Reeber and K. Wang [15].
c = a
0
The unstressed lattice parameter a0 is found using equation 7.16 and by replacing d0
in the regression formula.
0
c
a
N0
N0
30
22
)

d
2
2
N0
Polycrystalline and monocrystalline Model
2
N0
0 = a 0
(equ 7.18)
2
l
4
+ (h
3
⎛ c
⎜
⎝ a
⎞
⎟
⎠
2 ⎛ c
+ h k + k ) ⎜
⎝ a
The elastic behaviour of the buffer layer that is containing a small fraction of boron
nitride, about 3%, can be estimated by supposing a linear change of elastic constants
with increasing boron nitride content.
s
BGaN
ij
= s
GaN
ij
+ ( s
It is assumed that the compliance constants of the BGaN show the same thermal
behaviour like the GaN layer constants.
s ( T)
=
BGaN
ij
s(
T
)
GaN
ij
BN
ij
− s
s(
300
GaN
ij
s(
300 K)
The approximation of the temperature dependence was done by assuming that the
compliance constants will increase in the same way like the compliance constants of
gallium nitride. To estimate the new lattice parameters of BGaN the same procedure
) x
K)
N0
N0
BGaN
ij
GaN
ij
was applied like before, hence the value of x is again 3%.
And the new c0/a0 ratio is:
c
a
0
0
a
c
( T)
( T)
7.3. Strain evaluation
BGaN
0
BGaN
0
BGaN
BGaN
= a
= c
GaN
0
GaN
0
c0
=
a
0
+
+
( T)
( T)
( a
( c
GaN
GaN
BN
0
BN
0
− a
− c
0
GaN
0
GaN
0
) x
) x
c0
( 300 K)
a 0(
300 K)
c0
( 300 K)
a ( 300 K)
The strains are related to the sample system like in the stress evaluation. The only
strain that can be measured is expressed in the laboratory system. This strain is
parallel to G and so to eL3 vector of the laboratory system, thus the sample
coordinates must be expressed by the laboratory coordinates. The opposite way is
GaN
⎞
⎟
⎠
BGaN
BGaN
GaN
31

Polycrystalline and monocrystalline Model
expressed in equation 7.7, where the laboratory system is transformed into sample
system. For that reason the inverse tensor O D will be used to transform the strains
from the sample into the laboratory system.
ε = O O ε
ϕψ (equ 7.19)
ij
D
ki
The transposed matrix of O D is obtained by changing the suffixes. Like before the
measured strain in the laboratory system is written as:
ε
ϕψ
33
D
lj
ϕψ
d − d
=
d
The equation used in the nonlinear regression has the following form:
d
ϕψ
= d ( 1+
( ε
0
33
+ ( ε
22
+ ( ε
13
2
cos ( ϕ)
− ε
sin( ϕ)
− ε
12
23
0
0
kl
sin( 2 ϕ)
+ ε
11
cos( ϕ))
sin( 2 ψ)))
2
sin ( ϕ)
− ε
If the ω−axis is involved in the sample rotation the plane spacing is:
d
ϕω
= d ( 1+
( ε
0
33
+ ( ε
11
+ ( ε
2
cos ( ϕ)
+ ε
13
cos( ϕ)
+ ε
12
23
sin( 2 ϕ)
+ ε
22
sin( ϕ))
sin( 2 ω)))
33
2
sin ( ϕ)
− ε
33
2
) sin ( ψ)
2
) sin ( ω)
(equ 7.20)
(equ 7.21)
32

8. Deposition of Thin Films
Deposition of Thin Films
8.1. Magnetron sputtering of polycrystalline Al thin films on Si(1 0 0)
Magnetron sputtering of polycrystalline thin films is a widely used deposition
technique. Thin layers of chromium, gold, titanium, aluminium and many other
materials can be produced using the technique in a deposition chamber
schematically depicted in Fig. 8.1. In the evacuated chamber, ionized gas atoms
(usually argon ions) are accelerated under influence of an electrical field towards a
target. The high kinetic energy enables the ions to strike out surface bound atoms
which are leaving, with rather low kinetic energy (some keV), the target’s surface in
all possible directions. A substrate, fixed usually on the opposited side of the
chamber, is covered by a thin film of the sputtered atoms. The advantages of the
magnetron sputtering process are a good control of process parameters, high purity
of the films and a possibility to use low process temperatures.
For the production of polycrystalline Al thin films on monocrystalline Si(1 0 0) wafers,
magnetron sputtering device installed at the “Institute of Physics, Technical University
Wien” was used. As a substrate for the specimen preparation, a 1 mm thick Si(1 0 0)
wafer cleaned using isopropanol and acetone was used. On this native-oxidized
silicon wafer, a polycrystalline aluminium thin film with a thickness of 2 µm was
deposited using magnetron sputtering. The deposition was performed at 150 °C
applying a pressure of 4 x 10 -3 mbar.
+
Figure 8.1: A schematic view of a sputtering reactor
1 Vacuum chamber, 2 upper electrode, 3 plasma,
4 lower electrode, 5 gas inlet, 6 vacuum pump connection
7 substrate, 8 high frequency voltage, 9 target
10 plasma ion, 11 target atom
33

8.2. Molecular beam epitaxy of GaN/BGaN on Al2O3(0 0 0 1)
Deposition of Thin Films
The molecular beam epitaxy (MBE) is a thin film deposition process in which thermal
beams of atoms or molecules react on a single crystalline substrate surface that is
held under ultra high vacuum at elevated temperatures [25, 26].
Such devices usually consist of at least three chambers (Figure 8.2). The first
chamber serves as a sample container at medium high vacuum. The second
chamber is used for sample preparations such as outgassing or sputter etching and
for an accommodation of surface analytical facilities under ultra high vacuum. The
last chamber or growth chamber is also under ultrahigh vacuum and equipped with
facilities capable of forming and monitoring the vacuum level, heating and monitoring
the temperature of the substrate, generating and determining the molecular or atomic
beam’s intensity and controlling of the composition profiles.
The composition of the grown film and its doping level depend on the relative
deposition rates of the constituent elements, which is controlled by the evaporation
rate and the geometrical configuration between source and substrate. The
evaporation rate of particular elements is set by temperature of the effusion cells,
containing the element in the liquid phase.
MBE has in comparison with other epitaxial growth methods some unique
advantages. The growth rate is generally low allowing a compositional and doping
profile changes within atomic dimensions leading also to an atomic smooth surface.
The growth temperature is relatively low and thus diffusion between layers of different
composition is negligible. A disadvantage is due to low growth rates the small
throughput having an effect on long depositions times, about one hour for a one
micron high layer.
Figure 8.2: A schematic drawing of the growth chamber
For the purposes of this diploma thesis, the GaN/B0.03Ga0.97N multilayer structure
(with the individual thickness of 1 µm in the case of both sublayers) was prepared in
34

Deposition of Thin Films
the development laboratories of “Infineon AG, München”. In the preparation phase
the substrate is degreased and etched for the removal of surface contaminates and
mechanical damage due to polishing, and finally rinsed in deionised water. The
deposition of both sublayers on monocrystalline Al2O3(0 0 0 1) was performed at
700 °C in ultra-high vacuum gradually - at first BGaN sublayer was prepared and
then the deposition process was concluded with the deposition of GaN sublayer. The
original goal was to understand an influence of the BGaN buffer layer on the stress
state in the GaN top layer.
35

X-ray Diffraction – Measurements and Alignment
9. X-ray Diffraction – Measurements and Alignment
9.1. Four Circle Goniometer
The basic principle of characterising residual stress in thin layers with the help of a
diffractometer is to measure lattice spacing by using X-ray diffraction at different
sample tilts. The deviation from the unstressed lattice parameters in combination with
a material model enables the evaluation of the residual stresses. All present
measurements were carried out on a Seifert 3000 PTS four circle diffractometer
equipped with the DHS 900 heating stage.
The X-ray radiation is produced by rapid deceleration of electrons and can be
separated in continuous and characteristic radiation. In a diffraction experiment it is
desirable to have monochromatic Kα radiation. This is achieved by placing a filter
material, which has its K absorption edge between the Kα and Kβ wavelengths, in
the beam path. A common filter material for copper radiation is nickel [27].
Figure 9.1: Diffraction geometry
36

X-ray Diffraction – Measurements and Alignment
In order to obtain a convergent and parallel beam, a collimator is mounted on the X-
ray tube housing. The collimated beam irradiates the sample surface, where the
biggest part is absorbed and only a small part is diffracted, which can be detected by
a scintillation counter at certain θ-angles.
The collimator is replaced if the high resolution monchromator will be used. To
achieve a sufficient high intensity the disturbing filter material must be removed, so
the restriction of wavelengths is now guaranteed by the high resolution
monochromator, because only beams of a certain wavelength undergo two times
double diffraction.
The geometry of a diffraction experiment is sketched in figure 9.1, where the θ-angle
concerns the detector rotation and the three other angles ψ, ϕ and ω characterise the
sample tilt.
9.2. DHS 900 Domed Hot Stage
A stress measurement at elevated temperatures requires a devise capable of a
controlled heating of various thin film samples. In cooperation with the company
“Anton Paar”, the “Erich Schmid Institut für Materialwissenschaften“ and the
“Technische Universität Graz” the heating chamber DHS 900, Domed Hot stage, was
developed to fulfil important demands on elevated-temperature stress analysis.
The DHS 900 operates in a temperature range from 25°C up to 900°C controlled by
the TCU TEX temperature controller, where temperatures can be entered manually
or by a simple C-program. The advantage of setting the temperature via the serial
port is an automatic controlled temperature dependent stress measurements. The C-
Program can be seen in the appendix.
During the heating the reactivity of a sample increases with temperature. To protect
the layer from oxidation an inert gas or nitrogen is kept under a small dome. This
circumstance means that the primary beam and the scattered beams must go
through the dome, and no additional diffraction should be produced by the dome
itself. Thus organic material like PEEK is an excellent option, because of its small
absorption and scattering length, but the disadvantage of plastic is the low melting
temperature and therefore it is necessary to cool the dome with air. So the supply by
gas, air and electricity causes a limitation in rotation, thus it is recommended to stay
in a ϕ range from -100° to 50°.
37

9.3. The High-Resolution Monochromator
X-ray Diffraction – Measurements and Alignment
To measure residual stresses in a multilayered structure with very small differences
in lattice spacing, it is necessary to use a monochromatic radiation with a very narrow
wavelength (or energy) distribution. In this case, it is usual to use a high-resolution
monochomator
Figure 9.2: Monolithic monochromator with housing
Figure 9.2 shows a picture of the high resolution monochromator, produced by the
“Instituto dei Materiali per l’Elettronica ed il Magnetismo, Parma, Italy”. The design is
based on the Bartel’s monochromator, shown in figure 9.3, which is including two
channel-cut crystals cut out of a piece single crystalline germanium. In each crystal
two separate symmetric {2 2 0} diffractions occur, so the final beam is in the same
direction as the incident beam from the X-ray source.
Figure 9.3: Bartels monochromator with possible rotations for alignment
38

X-ray Diffraction – Measurements and Alignment
In the present case, the high resolution monochromator was manufactured out of a
piece of germanium single crystal, where the two channel cut crystals are joined by a
spring. This significantly simplifies the usually complicated alignment procedure.
The cut through the two blocks was done by taking into account two different
scattering planes. In the first block (1 1 1) and ( 1 1 1 ) and in the second block
( 2 2 0) and (2 2 0) planes are involved. The basic idea is that two couples of
reflection with non parallel scattering vectors are being obtained in the same crystal
for some specific geometry and by inducing a small bending δ of the crystal, as can
be seen in figure 9.4. The monolithic design makes it not necessary to tilt the
alignment like it is done with Bartels monochromator, since the crystal cut is designed
to guarantee that the scattering plane in the first and the second block coincide within
a good accuracy.
Figure 9.4: Monolithic monochromator under small bending
The final wavelength, dispersion and divergence are of interest for the measurement.
A way to qualitatively evaluate these parameters can be found by using DuMond
diagrams for multiple diffractions. According to the dynamical theory of X-ray
diffraction, the reflectivity of a perfect crystal for a fixed wavelength is close to 100%
in a finite ∆α range of several seconds of arc. Thus the DuMond diagram of a perfect
crystal is a band of width ∆α in the λ−α space. In the DuMond diagram, shown in
figure 9.5, two bands of wavelengths for several diffractions are plotted, one for the
first block with asymmetric (1 1 1) reflection and one for the second block with
symmetric (2 2 0) reflection. The cross section of the DuMond diagram for two
crystals gives the wavelength dispersion and the divergence of the final X-ray beam.
For the high resolution monochromator a wavelength dispersion ∆λ / λ = 2·10 -4 and a
divergence of ∆α = 15,4 arcsec is obtained, by using CuKα X-ray radiation.
39

X-ray Diffraction – Measurements and Alignment
Figure 9.5: DuMond diagram for of the the Ge(1 1 1, 1 1 1 )( 2 2 0, 2 2 0) setting
This device produces a diffracted beam with the minimum divergence in the
scattering plane but with the maximum divergence of several degrees in the
perpendicular direction, which does not affect the width of the diffraction profiles to be
measured if the scattering plane of the sample is aligned to the scattering plane of
the monochromator.
The final intensity of the high resolution monochromator is more than 50% higher
than that of the Bartels setting, thus the high resolution monochromator is a powerful
device for high resolution X-Ray diffraction.
40

9.4. Alignment of the diffractometer – point focus
X-ray Diffraction – Measurements and Alignment
To obtain reliable and reproducible results from X-ray diffraction stress
characterization in ψ-geometry, the beam must be directed to the intersection of
three rotation axes ω, ϕ and ψ, so that the beam spot is not allowed to move on the
sample surface during rotation or tilting. The alignment procedure is carried out for
the collimator with a point focus.
At the beginning the detector is set to 90° position an d the ω, ψ and ϕ angles are set
to 0°. Afterwards a laser is mounted on the shade hol der. If the laser is turned on, a
little red spot will appear on the diffractometer plate. Now a small fluorescent plate is
fixed on the diffractometer plate and the grid’s origin printed on the fluorescent plate
must be in the middle of the laser spot.
A dial gauge is fixed in front of the diffractometer plate to adjust the height until the
distance measured from the small fluorescent plate surface is 5,75 mm. After that the
dial gauge is removed.
The ϕ angle is set to 180° and the laser spot must stay in the origin of the grid. If not
the shade holder must be oriented, in such a way that the laser spot is back in the
middle. After turning back to ϕ = 0° the spot should stay in the origin.
The laser spot is now parallel to the ϕ-axis and, as next, the ψ angle is set to 85°.
The spot is distorted into a line, but the middle of the line must stay in the origin of the
grid. One more verification is made by setting ψ back to 0° and going with θ to 5°.
Again a line appears while its midpoint must coincide with the grid’s origin on the
fluorescent plate. After this procedure, the laser beam is directed to the intersection
of ω, ϕ and ψ axes (table 9.1).
θ ψ ω φ Note
90° 0° 0° 0° / 180° The laser spot must stay in the origin.
90° ± 85° 0° 0° The midpoint must stay in the origin.
5° 0° 0° 0° The midpoint must stay in the origin.
Table 9.1: Detector alignment
The next alignment procedure will concern the collimator (table 9.2). First the ψ angle
is set to 0°, the detector is driven to 155° and after wards the ω−angle must be in
90° position. If the shutter is opened a weak spot will appear on the fluorescent plate
surface. Like in the detector system alignment the middle of the spot must coincide
with the grid’s origin. If not, the shutter must be closed at first and then the collimator
will be adjusted. For that reason the whole procedure must be repeated several times
until the middle of the X-ray spot coincides with the grid’s origin.
41

X-ray Diffraction – Measurements and Alignment
The previous step was only a rough alignment. To be sure that the spot is in the
origin, ψ angle is set to 85°. The beam spot will appear as a li ne, and if the midpoint
does not match with the origin, the collimator has to be adjusted in vertical direction.
The alignment is nearly finished only the adjustment in the horizontal direction must
be done, thus ψ is changed to 0° and ω is set to 3°. Afterwards the procedure is done
in a similar way like in the vertical alignment.
At the end the small fluorescent plate is replaced by an annealed gold film. A θ-scan,
to measure the (4 2 0) peak with Cu-Kα radiation at a ψ-tilt of 0° and 60°, is
performed after the height adjustment, where the dial gauge must show a value of
5,75mm. The difference of the peak’s maxima at different ψ-tilts must not be more
than 0,015° and the small angular misalignment betwe en measurement value and
(4 2 0) peak position will be electronically corrected.
θ ψ ω φ Note
155° 0° 90° 0° The x-ray spot must be in the origin.
155° 85° 90° 0° The midpoint must be in the origin.
155° 0° 3° 0° The midpoint must be in the origin.
Table 9.2 Collimator alignment
9.5. Alignment of the diffractometer – line focus
In the case of the point focus alligment, the main aim was to find the intersection
point of all three rotation axes and focus the centre of the X-ray beam into this point.
Performing measurements with the high resolution monochromator in line focus
mode, the beam must just intersect the ω−rotation axis.
Before the X-ray beam alignment is done it is important to ensure that the Bragg
condition is fulfilled by the high resolution monochromator. Therefore all angles are
set to 0° and the secondary slit is removed so the whole detector area is used to
collect incoming photons. The tube’s voltage and current must be less than 20kV and
20mA to avoid a damage of the detector. By inducing a small bending on the
germanium crystal, the channel cut parts of the crystal are tilted a little bit, turning the
net planes into diffraction position. A maximum intensity of 80.000 to 90.000 counts
per second is a good standard value.
Then the shade holder with the laser is mounted and the θ-angle is set to 90°. A
cuboid shape like metallic block with parallel faces is fixed on the diffractometer plate,
42

X-ray Diffraction – Measurements and Alignment
so that the centre of the top face is in the middle of the laser spot. Afterwards the
shade holder together with the laser is removed and the detector is turned back to
0° position. Afterwards a dial gauge is mounted in fro nt of the diffractometer plate to
adjust the height to 5,75 mm.
The top surface of the metallic block intersects with the ω−axis. If no radiation is
detected by opened shutter the height of the monochromator is too low and all
radiation will be adsorbed by the metallic block (figure 9.6a). Otherwise if it is too high
the detector will measure the full intensity (figure 9.6b). To find the correct height the
monochromator is moved in such a way that half of the maximum intensity is
measured, so it is guaranteed that the X-ray beam intersects with the ω−axis
(figure 9.6c).
a.) Monochromator is too low b.) Monochromator is too high c.) X-ray beam intersects with
rotation axe
The fine adjustment is done by rotating the metal block around ω−axis. A scan shows
that the intensity has a maximum at a certain ω−angle (figure 9.7). It is supposed that
at the maximum intensity the top surface of the metallic block is parallel to the
incident beam, so the deviation from the zero ω−angle can be electronically
corrected. The last step is to measure gold standard like it is done in the point focus
alignment.
Figure 9.6: Height adjustment of the monochromator
43

I / counts s -1
83000
82000
81000
80000
79000
78000
77000
76000
75000
X-ray Diffraction – Measurements and Alignment
-0,4 -0,2 0,0 0,2 0,4
ω / °
Figure 9.7: Fine adjustment
44

10. Aluminium on silicon measurement
10.1. Experiment
Aluminium on silicon measurement
The aluminium layer, deposited by magnetron sputtering on silicon substrate, was
examined in a temperature controlled stress analysis performed on a four circle
diffractometer operating in point focus mode. The thermal load of the sample, which
included two thermal subcycles at elevated temperatures, was in a range from 25°C
to 450°C.
The temperature remained unaltered during the X-ray diffraction analysis and the
production of new measurement files by copying and by rewriting the θ-range. All
these processes took about 27 minutes together with the setting of the new
temperature followed by the beginning of the next X-ray diffraction analysis. The
order of temperature steps that were set in the experiment can be seen in figure 10.1.
At the beginning the specimen was heated up from room temperature to 275°C by a
step width of 25°C. The last temperature coincided wit h the beginning of the first
thermal subcycle, where the temperature was decreased step by step to a lower limit
of 150°C followed by an increase up to 375°C. During this heating the first thermal
subcycle ended at a temperature of 275°C.
T / °C
500
400
300
200
100
0
First thermal
subcycle
0 200 400 600 800 1000 1200
t / min
Second thermal
subcycle
Figure 10.1: Time table of the cycle experiment
45

Aluminium on silicon measurement
The last temperature was also the beginning of the second thermal subcycle that
caused the sample’s temperature to decrease by a step width of 25°C to the lowest
value of 250°C. Like at the lowest temperature in th e first subcycle, the sample was
heated up again to the maximum temperature of 450°C . At the end, the cooling of the
specimen was shortened by an increase of the step width to a value of 100°C.
10.2. Shift of diffraction peaks
The peak positions were changing during heating, caused by three effects, without
considering delamination, crack formation and precipitation or solution of phases at
elevated temperatures. The first effect is based on thermal or extrinsic stresses that
are strongly depending on the difference of thermal expansion coefficients between
film and substrate and/or between matrix and precipitations. The second effect is the
thermal expansion of the sample and the last effect is due to the changes in thin film
architecture by diffuse processes, like grain growth and recrystallisation. In figure
10.2 the change of peak position is shown for the (6 2 0) substrate reflection at a
ψ-angle of 0°.
I / counts s -1
60000
50000
40000
30000
20000
10000
0
127,0 127,2 127,4 127,6 127,8 128,0 128,2 128,4 128,6 128,8
2 θ / °
Figure 10.2: Peak shift of (6 2 0) substrate peak at ψ = 0°
300
250
200
150
100
50
0
T / °C
46

Aluminium on silicon measurement
The two peaks originated from identical Bragg reflection and were traced back to
characteristical copper Kα1 and Kα2 radiation. The peak located at lower θ-angles
was referred to copper Kα1 and the other one to copper Kα2 radiation.
The shown silicon peaks in figure 10.2 were measured in the first heating phase from
room temperature to 275°C. With increasing temperatur e the sample expanded, so
that the size of the lattice enlarged and therefore the reciprocal lattice spacing was
reduced.
I / counts s -1
In figure 10.3 the peak shift of the (3 3 1) aluminium reflection at ψ = 0° is shown in
the same temperature range as mentioned before, where the peaks shifted to lower
θ-angles due to the thermal expansion of the lattice in combination with residual
stresses.
300
250
200
150
100
110
111
112
2 θ / °
It is obvious, that the intensity of the layer peaks in comparison with the substrate
peaks was much lower because of the structural differences. The substrate is single
113
300
Figure 10.3: Peak shift of (3 3 1) aluminium peak at ψ = 0°
250
200
150
100
50
0
T / °C
47

Aluminium on silicon measurement
crystalline, so at a certain θ, ϕ and ψ position all net planes within the irradiated
volume were involved in the scattering process, in contrast to the polycrystalline
layer, where only grains in particular orientations were diffracting.
Another difference between layer and substrate is the peak shift which is reflected by
the thermal expansion coefficients of the silicon substrate, 2,6 10 -6 K -1 , and the thin
aluminium layer, 23,8 10 -6 K -1 . The mean peak shift obtained from the measurement
data was for the aluminium layer 0,159°K -1 and that of the substrate had a value of
0,0187°K -1 , so the ratio of thermal expansion coefficients and the ratio of peak shifts
are similar.
10.3. Peak broadening with increasing ψ tilt
During the stress analysis it was essential to tilt the sample by combination of two
angles, like it was done in the experiment by using ψ and ϕ-angle.
I / counts s -1
150
100
50
0
(3 3 1); ψ = 0°
(3 3 1); ψ = 30°
(3 3 1); ψ = 45°
(3 3 1); ψ = 60°
111 112 113 114
2 θ / °
Figure 10.4: Aluminium peaks at 25°C
Figure 10.4 shows (3 3 1) aluminium reflections of the first X-ray diffraction
measurement at room temperature for ϕ = 0° and at different ψ-tilts. Consider the
black curve, where a separation between Kα1 and Kα2 peak can be easily done. At a
48

Aluminium on silicon measurement
tilt of ψ = 30° the peak shape is becoming broader (red), theref ore both copper Kα
peaks start to merge, but one can still separate between Kα1 and Kα2 peak. The
peak profile at ψ = 45° (purple) is asymmetric and broader than the last two curves.
At the highest ψ-tilt, the peak profile (blue) seems to be symmetric, although it
consists of Kα1 and Kα2 radiation.
One reason for the peak broadening was the elongation of the beam spot caused by
sample tilt. Figure 10.5 shows the change of an originally circular beam spot at
different ψ and θ-angles. The region from where X-rays were diffracted enlarged with
increasing sample and decreasing θ-tilt. A rotation around the ψ-axis had the effect
that half of the sample surface was behind and half of it was in front of its original
plane. In these regions the beam was not longer exactly focused on the sample
surfaces, which led to a decreasing intensity simultaneously with a broadening of the
reflection profile.
Usually the integral intensity of the peaks remains constant, but the detector
equipped with a system of receiving slits, recorded only radiation diffracted from a
small, constant area of the sample surface, therefore the intensity decreased with
increasing sample tilt. It is worth to mention that such a beam spot distortion has no
effect on the θ-position of the peaks.
Figure 10.5: Distortion of beam spot
49

11. GaN and GaBN on sapphire measurement
GaN and GaBN on sapphire measurement
11.1. Measuring with the high resolution monochromator
For this measurement it should be noted that the GaN-layer, the BGaN buffer-layer
as well as the substrate are single crystalline, so a Bragg reflection can only be
obtained by rotating the sample in a certain position depending on the
crystallographic orientation. Therefore a couple of certain angles has to be used to
measure lattice spacing for each (h k l) reflection.
To differentiate between BGaN and GaN reflections, a high resolution X-ray
diffraction experiment was performed to resolve the small difference in-plane spacing
between GaN layer and BGaN buffer layer. For that reason the collimator was
replaced by the high resolution monochromator that operates in line focus mode.
In figure 11.1a the combination of ψ and ϕ-axes would distort the primary beam’s
projection and would cause a radiation of different specimen regions by varying the
sample tilt. On the other hand the ω - ϕ couple is more favourable, because there is
no influence on the spot shape, as can be seen in figure 11.1b.
a.) Psi – ϕ tilting b.) Omega - ϕ tilting
Figure 11.1: Choice of rotation axes and distortion of the primary beam’s projection
50

GaN and GaBN on sapphire measurement
It has to be noted that the ω−axis will always be parallel to the θ-axis if the
ω - ϕ couple is chosen. Thus the measured ω value is not the true tilt of the crystal
plane with respect to the sample system.
Figure 11.2: Definition of omega tilt
Figure 11.2 illustrates a small volume cut out of the sample. The thick black line
symbolizes a net plane where only a small part of incident beam is diffracted, by
showing an angular difference of 2θ between primary and scattered beam.
The measured ω−angle ωm is defined as angle between the sample surface and the
primary beam. However, a true ω value, which is demanded in the evaluation, is
represented as angle between net plane and surface.
2 θ
ω = − ωm
(equ 11.1)
2
In figure 11.2 it is obvious that the true ω can be calculated in equation 11.1 by
calculating the difference between θ and the measured ω−angle.
11.2. Stereographic projections and crystal orientation
Because there exists no simple procedure to estimate the find reflections in the
orientation space, the whole ϕ - ω space must be scanned to detect a certain Bragg
reflection. After two peaks were found, a calculation can be performed by using
numerical solvers to estimate the reflection that is perpendicular to the layer surface.
In the case of GaN and BGaN the (0 0 0 1) direction is parallel to the surface normal.
This information was used to generate a stereographic projection by the program
JWulf [28], that helped to find additional Bragg reflections by converting the true
51

GaN and GaBN on sapphire measurement
ω−angle, that is obtained by the stereographic projection, into the ω−value, that was
set in the measurement (equation 11.1). Due to the monochromatic wavelength and
the loss of intensity by the monochromator, it was necessary to perform scans
without shade, because the expected peaks had a low intensity and a low peak
width.
ω / °
22,0
21,5
21,0
20,5
20,0
19,5
19,0
18,5
The peak detection of a single crystalline material in a high resolution X-ray
diffraction measurement is a very sensitive process, where a small change of the tilt
angle ω can cause a peak to vanish as can be seen in figure 11.3. In the scan the
GaN peaks in ϕ−direction had a peak broadening of about 3,5° and the broadening in
ω−direction was about 0,5°.
-80 -60 -40 -20 0 20 40
A scan of the sapphire substrate can be seen in figure 11.4. The X-ray beam that
penetrates into the substrate was diffracted in a bigger volume than in the layer, so a
higher amount of atoms were taking part in the scattering process, which led to a
ϕ / °
Figure 11.3: {2 1 3 4} GaN Peaks
smaller peak width in comparison to the layer peaks.
50 counts s -1
100 counts s -1
150 counts s -1
200 counts s -1
250 counts s -1
52

GaN and GaBN on sapphire measurement
The substrate peak was of about 1° in ϕ-direction, and in ω−direction the broadening
was only about 0,2°. Since the peaks shifted with increasi ng temperature, their
position could sometimes be only detected by performing a scan without slit a small
ϕ - ω range.
ω / °
All Bragg peaks that were used to examine stresses in the thin film are plotted in the
stereographic projection, shown in figure 11.5. The surface normal of the sample
coincides with the (0 0 0 1) direction and the ( 1 0 1 0) direction is parallel to eS2
vector of the sample system if the misalignment of the sample in the DHS 900 is
neglected.
25,5
25,0
24,5
24,0
23,5
23,0
22,5
22,0
-80 -60 -40 -20 0 20 40
ϕ /°
0 counts s -1
20 counts s -1
40 counts s -1
60 counts s -1
80 counts s -1
Figure 11.4: {3 1 4 11} Sapphire peak
100 counts s -1
120 counts s -1
140 counts s -1
53

GaN and GaBN on sapphire measurement
eS2
Figure 11.5: Stereographic projection of GaN peaks
54
eS1

11.3. Phi adjustment
GaN and GaBN on sapphire measurement
Figure 11.3 and 11.4 show that the reflections were stretched in the ω - ϕ space, so
the main task was to find the peak maximum by varying ϕ and ω−angle. Before the
first ϕ-scan was carried out by setting an estimated ω and θ-angle, the shade holder
was removed to ensure that the whole detector area could be used to detect a peak
maxima. The ϕ-scan at room temperature is plotted in figure 11.6 where three peaks
belonging to the same family of net planes are shown.
I / counts s -1
300
200
100
0
ϕ (2 2 0 5) / °
18 20 22 24 26 28 30 32 34
400
-105 -100 -95 -90
ϕ (2 0 2 5) / °
-42 -40 -38 -36 -34 -32 -30 -28
ϕ (0 2 2 5) / °
Figure 11.6: Phi scans of {2 0 2 5} peaks
ϕ (2 0 2 5)
ϕ (0 2 2 5)
ϕ (2 2 0 5)
55

GaN and GaBN on sapphire measurement
The peak profiles in figure 11.6 were not fitted by a Gaussian peak profile. An
explanation of such a peak shape is given by the wide detector range measuring
without a shade. Figure 11.7 shows a peak that is going to be scanned where the
detector is assumed to be the reference system. For that reason the environment and
the peaks are moving. A peak that will come into the measuring range of the detector
increases the measured intensity until the whole peak is in the detector range, thus
the peak profile will show a plateau region.
So the peaks were not fitted by a simple Gaussian peak function that is unable to
describe the plateau region of the measured ϕ-peaks, but the evaluation can be done
by introducing the regression parameter c instead of the quadratic function in the
exponential term.
Figure 11.7: Peak shape distortion
I ( θ)
= I
0
⎛
⎜ ⎛ θ - θ
+ a exp - ⎜
⎜ ⎜
⎝ ⎝ b
0
c
⎞ ⎞
⎟ ⎟
⎟
⎠
⎟
⎠
56

11.4. Omega adjustment
GaN and GaBN on sapphire measurement
Like in chapter 11.3 all scans were carried out without a slit. In the ω−scans the new
ϕ-angle at maximum intensity was set and the former estimated θ-angle was kept.
Such scans, performed at room temperature, of {2 0 2 5} net planes are shown in
figure 11.8. The second peak cannot be traced back to copper Kα2 radiation
because of the restricted wavelength spectrum. Therefore it is supposed that the
peak at higher ω−angles refers to BGaN due to lower lattice constant caused by
substitution of gallium by boron atoms.
The ω−position of GaN and BGaN is assumed to be hardly influenced by the
ϕ-position because of the large peak width in ϕ-direction. Therefore ω−scans for both
layers were performed at same ϕ-tilt.
I / counts s -1
800
600
400
200
0
31,0 31,2 31,4 31,6 31,8 32,0
ω / °
Figure 11.8: Omega scans of {2 0 2 5} peaks
ω (2 0 2 5)
ω (0 2 2 5)
ω (2 2 0 5)
57

11.5. Theta scans:
GaN and GaBN on sapphire measurement
The θ-angle is required to calculate the plane spacing, the crucial parameter for
stress and strain evaluation. At first, the sample was tilted into the ω and ϕ-orientation
determined by the procedures from Sec. 11.3 and 11.4. It was necessary to
distinguish between GaN and BGaN θ-peaks with respect to their ω−positions. These
scans were performed using a 1mm slit shade. The results are presented in figure
11.9 for GaN and in figure 11.10 for BGaN.
I / counts s -1
600
500
400
300
200
100
0
135,8 136,0 136,2 136,4 136,6 136,8 137,0 137,2
2 θ / °
2θ (2 0 2 5)
2θ (0 2 2 5)
2θ (2 2 0 5)
Figure 11.9: Theta scans of {2 0 2 5} GaN peaks
58

I / counts s -1
800
600
400
200
GaN and GaBN on sapphire measurement
0
136,4 136,6 136,8 137,0 137,2 137,4 137,6 137,8
The same procedure was used to gain information about sapphire peaks as well. The
substrate peaks, like (0 0 0 12), showed much higher intensity but smaller peak width
than the layer peaks, so the detection was very difficult even the shade holder was
removed. The substrate peak shape had no Gaussian profile, as can be seen in
figure 11.11, because of the same reasons mentioned in chapter 11.3 or maybe
because of the dynamic theory of X-ray diffraction, where this peak could be
interpreted as Darwin curve.
2 θ / °
2θ (2 0 2 5)
2θ (0 2 2 5)
2θ (2 2 0 5)
Figure 11.10: Omega scans of {2 0 2 5} GaBN peaks
59

I (2 2 0 5) / counts s -1
500
400
300
200
100
2 θ (0 0 0 12) / °
2 θ (2 2 0 5) / °
GaN and GaBN on sapphire measurement
90,3 90,4 90,5 90,6 90,7 90,8 90,9
600
7000
2θ (0 0 0 12)
2θ (2 2 0 5)
0
0
135,6 135,8 136,0 136,2 136,4 136,6 136,8 137,0 137,2 137,4
6000
5000
4000
3000
2000
1000
Figure 11.11: Comparison between GaN layer and sapphire substrate peak
I (0 0 0 12) / counts s -1
60

12. Results of aluminium on silicon
12.1. Sin(ψ)² vs. a plot
Results of aluminium on silicon
The principle behind the stress evaluation is the projection of stresses defined in the
sample coordinate system on the measuring direction that is usually the eL3 axis of
the laboratory system. If the sample is not tilted around the ψ-axis the eS3 direction
will be parallel to the eL3 direction. For that reason no stresses can be evaluated if the
sample is in a biaxial stress state that has no components in eS3 direction.
σ33 ϕψ
= ( σ
11
2
sin ( ϕ)
− σ
12
sin( 2
ϕ)
+ σ
22
2
2
cos ( ϕ))
sin ( ψ)
Because of the isotropic behaviour of the thin Aluminium film, it is supposed that the
in-plane stresses (σ11, σ22) are equal, which leads to the non-diagonal elements
σ12 = 0.
a / A
4,080
4,075
4,070
4,065
4,060
4,055
4,050
4,045
4,040
275°C
250°C
225°C
200°C
175°C
150°C
125°C
100°C
75°C
25°C
0,0 0,2 0,4 0,6 0,8
sin 2 (ψ)
Figure 12.1: Lattice spacing dependence on sin² (ψ)
61

Results of aluminium on silicon
Such a simplified regression is mainly depending on the ψ-tilt and implies a linear
behaviour of a vs sin²(ψ). The measurement points, in figure 12.1, show the lattice
spacing at ψ = 0°, 30°, 45° and 60° evaluated from the 2- θ positions of the peaks,
starting from room temperature to 275°C.
If the sample is tilted the peak position will be influenced by residual stresses that are
responsible for the change of lattice spacing and the shift of peaks at constant
temperature. The slope value of the linear smoothing function provides information
about the stress state, so a positive sign indicates that the layer is under tensile
stress and if a negative sign is obtained then the opposite stress state will be present.
Furthermore the absolute value of the slope is proportional to the magnitude of
stress.
In figure 12.1 can be seen that the layer was in a tensile stress state at room
temperature. With raising temperature the slope is slightly decreasing and is nearly
zero at 125°C. A further heating was increasing the com pressive stress until 225°C
where the layer started to relax to lower compressive stress values.
a / A
4,080
4,075
4,070
4,065
4,060
4,055
4,050
275°C
250°C
225°C
200°C
175°C
275°C
250°C
225°C
200°C
175°C
150°C
0,0 0,2 0,4 0,6 0,8
sin 2 (ψ)
Figure 12.2: First thermal subcycle
Heating
Cooling
62

Results of aluminium on silicon
In the first thermal subcycle the temperature was changed in steps of 25°C from
275°C to 150°C, followed by an increase of temperatur e up to 275°C. In figure 12.2
the stresses increased from compression to tension during the cooling, which is
expressed by the blue lines. The approximate border between the two stress states
was found at 225°C.
The heating, symbolised by the black lines, reduced the tension and turned the layer
back into a compressive stress state at a temperature of about 225°C. It can be seen
that there is nearly no difference in stress behaviour between heating and cooling
operation because the linear smoothing functions are nearly parallel.
The dependence of the lattice spacing on the sample tilt in the second thermal
subcycle, which is shown in figure 12.3, was measured in a temperature range from
250°C to 375°C with a step width of again 25°C. In t he first X-ray analysis of the
second subcycle at a temperature of 375°C the specimen was under compressive
stresses. The followed cooling increased the stress value until 325°C, where a further
reduction of heat did not seem to affect the stress of the following temperature steps.
a / A
4,090
4,085
4,080
4,075
4,070
4,065
375°C
350°C
325°C
300°C
275°C
250°C
375°C
350°C
325°C
300°C
275°C
0,0 0,2 0,4 0,6 0,8
sin 2 (ψ)
Figure 12.3: Second thermal subcycle
Heating
Cooling
63

Results of aluminium on silicon
Tension was dominating at the lowest temperature of 250°C. A followed heating
enlarged the lattice spacing and decreases the slope of the smoothing function, so
the stress changed back to compression at 300°C.
In the cooling operation the border between tension and compression was detected
at about 350°C and in the heating operation it was f ound about 50°C lower than
before. At the end of the second thermal subcycle the black and blue linear
smoothing functions are nearly at same position.
12.2. Lattice spacing of aluminium layer
In equation 7.11 not only the in-plane stress but also the mean unstressed lattice
parameter, of all cubic face centred aluminium crystals that are present in the
irradiated sample volume, were evaluated.
a 0 / A
4,10
4,09
4,08
4,07
4,06
4,05
4,04
0 100 200 300 400 500
T / °C
a 0 = a + b T + c T 2
a = 4,0488
b = 6,8827 10 -5
c = 7,4653 10 -8
Figure 12.4: Thermal expansion of aluminium lattice spacing
The thermal expansion of the lattice spacing a0 of the thin aluminium layer is plotted
in figure 12.4 and shows only weak scattering of the data. An exponential function
proved to be favourable to fit the temperature dependent lattice spacing.
64

12.3. Thermal expansion coefficient of aluminium
Results of aluminium on silicon
The thermal expansion coefficient of the aluminium layer can be calculated [29] by
entering the temperature dependent fit function, which expresses the unstressed
lattice spacing of the previous paragraph, in equation 12.1.
1 ∂ a 0(
T)
α ( T)
=
(equ. 12.1)
a ( T)
∂T
0
The result of this application is plotted in figure 12.5. A comparison between values
from literature (chapter 6.1.1) and the calculated expansion shows a difference of
20% at room temperature.
α Al / 10 -6 K -1
34
32
30
28
26
24
22
20
18
16
0 100 200 300 400 500
T / °C
Figure 12.5: Thermal expansion coefficient of aluminium
65

12.4. Lattice spacing of silicon substrate
Results of aluminium on silicon
The lattice spacing of silicon substrate was evaluated by making use of the (6 2 0)
reflection. It is assumed that the thermal stresses of the substrate are negligible with
respect to relatively deep penetration depth of the X-ray beam, so the measurements
can provide an information about the thermal expansion of the substrate.
a 0 / A
5,440
5,438
5,436
5,434
5,432
5,430
12.5. Thermal expansion coefficient of silicon substrate
The procedure of calculating the thermal expansion coefficient of the silicon substrate
is done in the same manner as it was mentioned in chapter 12.3. The result can be
seen in figure 12.7, where the thermal expansion coefficient at room temperature is
2,43 10 -6 K -1 .
0 100 200 300 400 500
T / °C
a 0 = a + b T + c T 2
a = 5,4308
b = 1,2380 10 -5
c = 1,2058 10 -8
Figure 12.6: Thermal expansion of silicon lattice spacing
66

α Si / 10 -6 K -1
4,5
4,0
3,5
3,0
2,5
2,0
12.6. Stress curve
Results of aluminium on silicon
0 100 200 300 400 500
Owing to the heating and cooling operations, it is necessary to differ between these
two temperature gradients. Consequently the stress dependence of the heating is
drawn using black lines and the blue lines symbolize the stress dependence during
the cooling in figure 12.8.
T / °C
Figure 12.7: Thermal expansion coefficient of silicon substrate
At 25°C the layer was in a tensile stress state of 185 M Pa, but with raising
temperature the stress decreased linearly by 1,702 MPa·K -1 (12.8-1) until the
temperature reached 200°C where the linearity was lost (12.8-2) and the curve
shows a minimum at 225°C. At a temperature of 275°C the first thermal subcycle
was commenced. The linear stress-temperature dependences of the heating and
cooling are very similar which is reflected by the slope values of 1,957 MPa·K -1 of the
cooling operation and 1,995 MPa·K -1 of the heating operation (12.8-3 and 12.8-4).
67

Results of aluminium on silicon
After the first subcycle, the residual stress was in range between -70MPa and -
40MPa that was measured in a temperature interval from 275°C to 375°C.
The specimen temperature of 375°C was also the beginnin g of the second thermal
subcycle that started with a cooling operation (12.8-5). From 375°C to 325°C the
stress increased linearly with a slope of 1,581 MPa·K -1 , afterwards the slope changed
to 0,663 MPa·K -1 thus the stress increased less than before, but the dependence was
still linear. At the lower limit of the second thermal subcycle at 250°C, the sample
was heated again. During the heating the stress decreased with 1,876 MPa·K -1 until a
temperature of 325°C was reached (12.8-6). Afterwards the stress remained in a
range between -55MPa and -40MPa. At the end the sample was cooled to room
temperature with a step width of 100°C between the X -ray diffraction measurements.
It is supposed that the temperature dependent stress of the last cooling operation
(12.8-8) can be described by a linear behaviour with a slope value of 0,815 MPa·K -1 .
σ 11 / MPa
400
300
200
100
0
-100
-200
1
4
3
0 100 200 300 400 500
T / °C
Heating
Cooling
Figure 12.8: In-plane stress in aluminium layer
1 First heating phase, 2 deviation from linear behaviour,
3 cooling operation of first thermal subcycle, 4 heating operation of first thermal subcycle,
5 cooling operation of second thermal subcycle, 6 heating operation of second thermal subcycle,
7 decrease of compressive yield stress, 8 last cooling operation
2
6
8
5
7
68

12.7. Strain curves
Results of aluminium on silicon
The evaluated stress from chapter 12.6 is based on the strain measurement, since
the shift of Bragg reflections is referred to the change of plane spacing, which is
proportional to the change of strain ε33 in laboratory system.
In the chapter 12.1, the in-plane stresses σ11 and σ22 were assumed to be equal, and
the shear stress was neglected, so the strain evaluation must be based on this
assumptions. In the evaluation of the strain data use has been made of the
unstressed lattice spacing a0 received from the previous stress evaluation. The
reason why not calculating a0 in the strain evaluation is that it was done before, and
therefore more measurement points per regression parameter can be used in the
nonlinear regression. The results are shown in figure 12.9 and 12.10 for the in-plane
and out of plane strain-temperature dependence.
ε 11
0,004
0,003
0,002
0,001
0,000
-0,001
-0,002
0 100 200 300 400 500
T / °C
Figure 12.9: In-plane strain of aluminium layer
Heating
Cooling
69

ε 33
0,002
0,000
-0,002
-0,004
12.8. Discussion
Results of aluminium on silicon
0 100 200 300 400 500
T / °C
Figure 12.10: Out of plane strain of aluminium layer
During the deposition of the thin film, target atoms were adsorbed on the layer
surface transferring a part of their kinetic energy to the layer and the substrate, where
this additional energy is converted into heat and plastic deformation of the layer. The
rather low deposition temperature of 150°C, that was kept constant by a temperature
control devise during the film formation, led to a high supercooling accompanied by a
high density of nuclei. Owing to the low temperature diffusion was limited, and so
grain growth was negligible, resulting in a fine-grained film structure.
The deposition process was followed by a cooling operation that caused a
contraction of the layer as well as the substrate. The prevention of free contraction by
the substrate during cooling down was responsible for extrinsic in-plane tensile
stresses in the aluminium layer, because of the ten times higher thermal expansion
coefficient of aluminium. For that reason tensile in-plane stresses were obtained in
the first X-ray diffraction experiment at room temperature.
Heating
Cooling
70

Results of aluminium on silicon
The measured temperature dependent stress behaviour is shown in figure 12.8,
where the stress/temperature dependence of the first heating operation from room
temperature to 200°C is elastic. Only the extrinsic in-p lane stresses, but no additional
plastic deformation, are superimposed on the intrinsic in-plane stress state (12.8-1).
Therefore, the intrinsic in-plane stress value of -39MPa [30, 31] was derived at
150°C, which corresponds to the deposition temperature where no extrinsic stresses
were assumed to be present. The compressive intrinsic in-plane stress state was
traced back to atomic peening caused by the fabrication technique.
In the present case diffuse processes were activated at elevated temperatures
[6, 32], where at 200°C a deviation from the linear elastic behaviour was observed
(12.8-2). This nonlinearity is related to plastic deformation, grain growth and
recovery. According to F. Vollertsen and S. Vogler [32] dynamic recrystallisation of
aluminium can be excluded, because a high dislocation density is rapidly reduced by
recovery processes at elevated temperatures.
As mentioned before, the deposited layer formed as fine-grained structure, so the
reduction of interface boundary that is synonymous with a reduction of the integral
surface energy is the driving force behind the coarsening process [32]. The grain
growth will be inhibited if the grains reach a dimension which is in the order of the
layer thickness, and so it is assumed that the grain growth process ended at a
temperature of 275°C [6]. Such a coarsened structure m ay be regarded as bamboo
structure which represents an obstacle for electro migration.
The material responded in an elastic manner during the first thermal subcycle (12.8-3
and 12.8-4). A completely different behaviour was measured in the second thermal
subcycle which showed a hysteresis. The stress elastically increased starting at
375°C and -40,3MPa until 325°C, where the sample sta rted to yield again at a tensile
stress value of 38,5MPa (12.8-5). At 250°C the sample w as heated again and an
elastic behaviour was obtained up to 325°C (12.8-6). It can be seen that the
compressive yield stress was decreasing in the temperature range from 275°C to
450°C (12.8-7). This is a well known phenomenon. Plasti c deformation is described
through dislocation movement and diffusion driven processes. Those processes are
temperature dependent and cause a material to yield at lower stress values than at
room temperature.
Regarding to figure 12.8, it is obvious that the elastic behaviour of the cooling
operation in the second thermal subcycle (12.8-5) was measured within a
temperature interval of 325°C to 375°C, which is small er than the elastic response of
the heating operation (12.8-6) that lay in a temperature interval of 250°C to 325°C. If
the plastic yield stress of the cooling operation in the second thermal subcycle is
extrapolated to higher temperatures values, like it is done in figure 12.11, than the
extrapolated yield stress will intersect with the compressive yield stress of the cycle
(12.8-7) at a temperature lower than 450°C. Conseque ntly it is questionable if an
71

Results of aluminium on silicon
elastic behaviour is obtained during the last cooling operation (12.8-8) which cannot
be shown by the experimental data because of the rough step width of 100°C.
However, an elastic behaviour would have only little influence on the in-plane
stresses of the last cooling operation starting from 450° to room temperature that is
dominated by plastic deformation (12.8-8).
σ 1 / MPa
200
100
0
-100
-200
In figure 12.12 the decrease of intensity as a function of temperature is shown. The
data is scattered but an exponential decay is obvious. Of interest are the points
belonging to the last cooling steps (350°C, 250°C, 150 °C and 25°C), which lie close
to the dotted line.
250 300 350 400 450
T / °C
Figure 12.11: Extrapolation of plastic yield stress
Heating
Cooling
An explanation of the deviation of the exponential decay could be given by
considering the elongation of grains, caused by plastic deformation accompanied by
a preferred orientation of crystallographic planes. For that reason the lower intensity
values may be traced back to a change of texture during the last cooling operation.
72

I / counts s -1
180
160
140
120
100
80
60
40
T / °C
Results of aluminium on silicon
0 100 200 300 400 500
Figure 12.12: Decrease of intensity
(3 3 1); ψ = 0°
73

13. Results of GaN/GaBN/Al2O3(0 0 0 1)
13.1. Sapphire lattice parameters
Results of GaN/GaBN/Al2O3(0 0 0 1)
Temperature dependencies of sapphire lattice parameters were evaluated by
measuring (0 0 0 12) and (3 1 4 11) reflections of the substrate. Since the
penetration depth of the X-ray beam is much higher than the GaN/BGaN film
thickness, it can be supposed that the lattice spacings evaluated from the measured
data represent unstressed parameters.
The lattice parameters can be calculated using equation 7.16 and by entering the
calculated spacing from the Bragg equation 3.8. The temperature dependent lattice
parameters are plotted in Figure 13.1 and were fitted by a quadratic function.
c 0 / A
13,06
13,04
13,02
13,00
12,98
12,96
c 0
a 0
a 0 = a + b T + c T 2
a = 4,7581
b = 2,9193 10 -5
c = 7,5746 10 -9
100 200 300 400 500 600
T / °C
c 0 = a + b T + c T 2
a = 12,9849
b = 9,6315 10 -5
c = 1,5527 10 -8
Figure 13.1: Temperature dependencies of sapphire lattice parameters.
4,790
4,785
4,780
4,775
4,770
4,765
4,760
4,755
74
a 0 / A

13.2. Thermal expansion coefficients of sapphire
Results of GaN/GaBN/Al2O3(0 0 0 1)
The thermal expansion coefficients of the sapphire were evaluated from the functions
describing temperature dependencies of the lattice spacings using equation 12.1.
The results are plotted in figure 13.2, where the evaluated thermal expansion
coefficients show good correlation to literature values.
α / 10 -6 K -1
9,0
8,5
8,0
7,5
7,0
6,5
6,0
100 200 300 400 500 600
T / °C
Figure 13.2: Thermal expansion coefficient of sapphire
13.3. In-plane stress in GaN and GaBN layer:
The stress evaluation in GaB and in BGaN is based on the assumption that single
crystalline hexagonal thin films are oriented with (0 0 0 1) parallel to the sample’s
surface normal as described in chapter 7.2. Considering these specific geometrical
conditions, it can be supposed that the in-plane stress is isotropic (σ11 = σ22) while
surface normal stress components σi3 = 0. As a consequence of the assumed stress
state follows that the non-diagonal components σ12 = 0.
75

Results of GaN/GaBN/Al2O3(0 0 0 1)
The unstressed lattice spacing a0 and c0 of the hexagonal crystals were evaluated
using the equation 7.18. It is assumed that the ratio of the unstressed lattice
parameters a0/c0 is corresponding to the literature values measured by K. Wang and
R.R. Reeber [15]. The evaluated unstressed lattice parameter a0 is plotted in
figure 13.3.
a 0 / A
3,200
3,198
3,196
3,194
3,192
3,190
3,188
3,186
The in-plane stresses were measured in a temperature range from 50°C to 600°C
with a step width of 50°C. They behave in a nearly l inear way in the lower
temperature region, as can be seen in figure 13.4. In the beginning the in-plane
stress was in a compressive state of 440 MPa that is further reduced during the
heating cycle. By approaching the end of the heating procedure the thin film was in
an unstressed state.
0 100 200 300 400 500 600
T / °C
Figure 13.3: Measured lattice parameter a0 of GaN
76

σ 11 / GPa
0,0
-0,1
-0,2
-0,3
-0,4
-0,5
T / °C
Results of GaN/GaBN/Al2O3(0 0 0 1)
0 100 200 300 400 500 600
Figure 13.4: Inplane stress of GaN layer
The properties of BGaN sublayer were evaluated using the same approach as those
of the GaN top layer. Both have wurtzite structure and same orientation with respect
to the substrate. For the BGaN sublayer, the unstressed lattice parameter as well as
the in-plane stresses were evaluated. The unstressed lattice spacing ao is slightly
smaller than the ao spacing of the GaN layer, as can be seen in figure 13.5.
77

a 0 / A
3,194
3,192
3,190
3,188
3,186
3,184
3,182
3,180
Results of GaN/GaBN/Al2O3(0 0 0 1)
0 100 200 300 400 500 600
Figure 13.6 shows the in-plane stress of the buffer layer containing about 3% boron
nitride that is at 50°C under tensile stress of 350MPa. An increase of temperature
was attended by an increase of in-plane stress, for that reason the in-plane stress at
600°C has a value of 710MPa.
T / °C
Figure 13.5: Unstressed lattice parameter a0 of the GaBN buffer layer
78

σ 11 / GPa
0,8
0,7
0,6
0,5
0,4
0,3
Results of GaN/GaBN/Al2O3(0 0 0 1)
0 100 200 300 400 500 600
In figure 13.7 both in-plane and out of plane strain of the GaN top layer are
approaching an unstrained state with raising sample temperature. The in-plane strain
of the buffer layer in figure 13.8 is in a tensile state forcing the out of plane stress into
compressive strain regions.
T / °C
Figure 13.6: Inplane stress of GaBN buffer layer
79

ε 11
ε 11
0,0000
-0,0002
-0,0004
-0,0006
-0,0008
-0,0010
0,0016
0,0014
0,0012
0,0010
0,0008
0,0006
0,0004
T / °C
Figure 13.7: In and out of plane strains of GaN layer
Results of GaN/GaBN/Al2O3(0 0 0 1)
0 100 200 300 400 500 600
0 100 200 300 400 500 600
T / °C
Figure 13.8: In and out of plane strains of BGaN buffer layer
ε 11
ε 33
ε 11
ε 33
5e-4
4e-4
3e-4
2e-4
1e-4
0
-0,0002
-0,0003
-0,0004
-0,0005
-0,0006
-0,0007
ε 33
ε 33
80

13.4. Discussion:
Results of GaN/GaBN/Al2O3(0 0 0 1)
Sapphire is no perfectly suitable substrate for the growth of GaN layer due to the
enormous lattice mismatch. For that reason, attempts are made to reduce the
mismatch by introducing a buffer layer to minimize the defect density which would
degrade the GaN layer’s quality.
In the present case a GaN buffer layer containing about 3% boron nitride was
deposited on the sapphire substrate. If the anion sublattices of the sapphire and the
BGaN are continuous than the deposited buffer layer should be under compressive
stress [33], which cannot be shown by the experimental data.
The simple approach, by only considering geometrical aspects, is not sufficient to
explain the transition region between sapphire and GaN buffer layer that is
influencing the intrinsic stresses. Not only the structural difference between GaN and
sapphire that seems to be responsible for a tensile stress state, caused by island
coalescence [34] of the growing GaN layer and surface tension, but also the
substitution of GaN by BN, which has a smaller unit cell than GaN, contributes to
tensile stress in the buffer layer. An extrapolation of buffer layer’s in-plane
stresses/temperature dependence shows a value of 766MPa at the deposition
temperature at 700°C. The tensile stresses of the buffe r layer affects the GaN layer
that is, if the smoothing function is extrapolated, in a tensile stress state of 128MPa at
the deposition temperature. Therefore the intrinsic stresses of the buffer and GaN
layer are supposed to have a value of 766MPa and 128MPa, respectively.
Extrinsic stresses do not exist at a temperature of 700°C, because they can only
originate from a heating or cooling after the thin film formation. A cooling procedure
starting from the deposition temperature will cause a higher contraction of the
substrate than of the layer materials. For that reason the substrate will induce
compressive in-plane stresses in the BGaN and GaN layer, as is observed in the
experiment where the stresses at room temperature were much lower than at
elevated temperatures.
High resolution X-ray diffraction stress measurements were performed during the
cooling to provide information about an eventually plastic flow of the material.
However, the results showed a good correspondence between heating and cooling of
the sample, so it is assumed that the sample is only elastically deformed.
81

14. Conclusion and Outlook
Conclusion and Outlook
Elevated-temperature X-ray diffraction has been applied to evaluate temperature
dependencies of residual stresses in aluminium thin film on silicon (0 0 1) and in
sublayers of BGaN/GaN structure on sapphire. The results indicate that the method
can provide useful information with respect to the elastic and plastic response of the
(sub)layers, resolve residual stresses in sublayers with very small differences in
composition, extrapolate intrinsic and extrinsic stresses, observe phenomena related
to the changes of thin film architecture etc. The part of the data has been published
and submitted [31, 35]
Further studies on this topic should focus the stress relaxation in transition region
between plastic and elastic deformation that is attributed to grain growth of the
polycrystalline aluminium. A peak profile analysis coupled with scanning electron
microscopy measurements should provide an information about the coarsening
process of the aluminium layer in future examinations.
The weak textured layer was treated in an isotropic way during the evaluation. This
assumption cannot be applied on all polycrystalline thin films due to the circumstance
that most layers show a strong texture. The isotropic Hill model in chapter 7.1 can be
seen as groundwork for a future based ODF dependent formula capable of serving
as basis for a nonlinear stress evaluation of textured layer materials.
The sign and magnitude of the GaN layer in-plane stresses is an important issue in
the manufacturing process. For that reason a buffer layer is usually deposited on the
substrate, in order to provide a suitable lattice for the GaN film. The qualitiy of the
buffer lattice is influenced by residual stresses that are supporting defect formation
and finally leading to a degradation of the BGaN and GaN layer.
It was demonstrated that a separation of Bragg reflections belonging to different
materials with nearly same lattice parameters can be done by a restriction of the
wavelength spectrum using the high resolution monochromator. Further examinations
of GaN/BGaN/Sapphire multilayered structures should be performed to characterise
the in-plane stresses/temperature behaviour depending on the boron nitride content
of the buffer layer, so that the boron nitride influence on the temperature dependent
stresses can be resolved. Such results can serve in a future development of GaN
based applications.
82

15. Literature
[1] C. Kittel
Einführung in die Festkörperphysik
R. Oldenbourg Verlag, 1999
[2] P. Chadwick
Continuum mechanics, concise theory and problems
Dover publications, 1999
[3] J. F. Nye
Properties of crystals, their representation by tensor and matrices
Oxford university press, 2000
[4] W. Voigt
Lehrbuch der Kristallphysik
B. G. Teubners Lehrbücher, 1910
[5] K. O’Donell, J. Kostetsky, R. S. Post
Controlling stress in thin films
IMAPS Flipchips, Wilmington, 2002
[6] W. D. Nix
Mechanical properties of thin films
Metallurgical transactions A, Vol 20A., pp 2217, November 1989
[7] Bronstein, Semendjajew, Musiol, Mühlig
Taschenbuch der Mathematik
Verlag Harri Deutsch, 1995
[8] V. Randle, O. Engler
Introduction to texture analysis, macrotexture, microtexture & Orientation
Mapping
Gordon and Breach Science Publishers, 2000
[9] Hermann Schumann
Metallographie, 13. neu bearbeitete Auflage
Deutscher Verlag für Grundstoffindustrie Stuttgart, 1991
Literature
83

[10] G. Harsch, R. Schmidt
Kristallgeometrie, Packungen und Symmetrie in Stereodarstellungen
Diesterweg/Salle, 1993
[11] Landolt Börnstein
Literature
Elastische, piezoelektrische, piezooptische und elektrooptische Konstanten
von Kristallen
Springer Verlag Berlin – Heidelberg – New York, 1966
[12] Horst Kuchling
Taschenbuch der Physik
Fachbuchverlag Leipzig, 1996
[13] Micheal H. Jones, Stephan H. Jones
Basic Mechanical and Thermal Properties of Silicon
www.virginiasemi.com
[14] R. R. Reeber, K. Wang
High Temperature Elastic Constant Prediction of Some Group III-Nitrides
MRS Internet Journal Nitride Semiconductor Research Vol. 6, 2001
[15] K. Wang, R.R. Reeber
Thermal Expansion of GaN and AlN
Nitride Semiconductor Symposium 863-8, 1998
[16] J. I. Pankove, T. D. Moustakas
Gallium nitride (GaN) I
Semiconductors and semimetals, volume 50, 1998
[17] Anna E. McHale
Phase Equilibria Diagrams / Phase diagrams for ceramicists
Volume X, Figures 8665-9114
The American Ceramic Society, Inc. 1994
[18] K. Kim, W.R.L. Lambrecht, B. Segall
Elastic constants and related properties of tetrahedrally bonded BN, AlN, GaN,
and InN.
Physical Review B 53, S. 16310, 1996
84

[19] Y. M. Chiang, W. D. Kingery, D. Birnie III
Physical ceramics, principles for ceramic science and engineering
Wiley, 1997
[20] MarkeTech International
Sapphire Table of General Properties
www.mkt-intl.com
[21] Landolt-Börnstein
Numerical Data and Functional Relationship in Science and Technology
Crystal and solid Physics, Vol. 17, Semiconductors, Springer New York.
[22] P. van Houtte, L. de Buyser
The influence of crystallographic texture on diffraction measurement of
residual stress
Acta Metallurgica et Materialia, Vol. 41, No. 2, pp 323, 1991
[23] T. Uchida, N. Funamori, T. Yagi
Lattice strains in crystals under uniaxial stress field
Journal of applied physics, Vol 80, No. 2, pp 739, 1996
[24] J. Keckes, J. W. Gerlach, B. Rauschenbach
Literature
Residual stresses in cubic and hexagonal GaN grown on sapphire using ion
beam-assisted deposition
Journal of crystal growth 219 1-9, 2000
[25] J. I. Pankove, T. D. Moustakas
Gallium nitride (GaN) II
Semiconductors and semimetals, volume 57, 1999
[26] G. Koblmüller
Molecular beam epitaxy of group III-nitrides on silicon carbide
Diploma thesis TU Wien, 2001
[27] I. C. Noyan, J. B. Cohen
Residual stress, measurement by diffraction and interpretation
Springer Verlag, 1986
85

[28] Steffen Weber
JWulf
http://jcrystal.com/steffenweber/java.html
[29] P. A. Tipler
Physik
Spektrum Akademischer Verlag, Heidelberg – Berlin – Oxford, 1995
[30] E. Eiper
Literature
Einfluss der Temperatur auf mechanische Spannungen in dünnen, auf Silizium
gewachsenen Al-Filmen
Diplomarbeit TU Graz, 2003
[31] E. Eiper, R. Resel, et al.
Thermally-induced stresses in thin aluminium layers grown on silicon
Powder Diffraction 19(1), March 2004
[32] F. Vollertsen, S. Vogler
Werkstoffeigenschaft und Mikrostruktur
Hanser Verlag, 1989
[33] E. V. Etzkorn, D. R. Clarke
Cracking of GaN films
Journal of Applied Physics, Vol. 89, Nr. 2, pp 1025, 2000
[34] K. A. Dunn, S.E Babcock, D. S. Stone, et al.
Dislocation Arrangement in a thick LEO GaN Film on Sapphire
MRS Internet Journal Nitride Semiconductor Research 5S1, W2.11, 2000
[35] J. Keckes, M. Hafok, E. Eiper, et al.
Evaluation of experimental stress-strain dependence from thermally cycled Al
thin film on Si
Acta Materialica
86

16. Appendix
16.1. Serial port communication
#include
#include
#include
#include
#include
void main(int argc, char *argv[])
{
FILE *fschr;
unsigned int i, bcc, zeit;
char temp[6],zeich[0];
}
strcpy(temp,argv[1]);
bcc= 'S'^'L';
for (i=0;i

16.2. Stereographic projection of Silicon (0 0 1)
Appendix
88

16.3. Stereographic projection of gallium nitride (0 0 1)
Appendix
89

16.4. Stereographic projection of sapphire (0 0 1)
Appendix
90