Panalytical Ga NXRDbooklet PKidd
Panalytical Ga NXRDbooklet PKidd
Panalytical Ga NXRDbooklet PKidd
P. Kidd
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Table of contents
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
3. Principles of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Bragg’s law and measurement of Bragg peaks . . . . . . . . . . . . . . . . . . . . 69
3.1.1 Measurement in a diffractometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.2 Bragg’s law depicted in real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1.3 Bragg’s law depicted in reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.4 Alignment of azimuth into the diffraction plane . . . . . . . . . . . . . . . . . . 73
3.1.4.1 Major azimuths in (0001) oriented GaN layers . . . . . . . . . . . . . . . . . . . . . 73
3.1.5 Accessible reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.1.6 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.6.1 Scan units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Preface
At the time of writing, gallium nitride, GaN, and related compounds are emerging
as major players in the business of electronics. This book provides an introduction
to the X-ray diffraction analysis of key structural parameters in epitaxial GaN layers.
Fundamental crystallographic concepts are introduced and related to the specific
requirements of the technological structures created for optoelectronic and electronic
devices employing GaN and related compounds.
The book is written for those who wish to gain a deeper appreciation of the principles
behind XRD applications in the ever advancing field of compound semiconductor
devices. It is also written for those who are engaged in the measurement of epitaxial
layers who are looking for a background explanation of the concepts employed in
proprietary analysis software such as X’Pert Epitaxy, and those looking for a pathway
through the analysis process so that they can design their own calculations. Whilst
the solutions and the examples presented here are focused on GaN technology, the
principles are generally transferable to solutions for other compound semiconductor
systems.
The solutions discussed in this booklet apply to results from high resolution rocking
curves and reciprocal space maps. These methods are not the only methods used to
investigate advanced materials although they are the most common applications in
GaN device technology. Other X-ray diffraction methods, often used in a research
environment, include reflectometry, in-plane scattering, SAXS and GISAXS. New
methods are emerging to assist rapid measurement and automation. The details of
the experimental methods will be considered in a separate publication. All of the
measurements referred to in the booklet can be carried out on PANalytical X’Pert MRD
and X’Pert MRD XL equipment.
This booklet describes the precise and quantitative measurements that can be made
on epitaxial device structures. A number of alternative methods can also be employed
to measure the relative quality of defective epitaxial layers. Such approaches include
peak shape and width measurements and via this route an exploration of mosaic block
sizes, crystallographic tilts and rotations and estimations of defect densities and textural
spread. The principles involved in qualitative analyses are also introduced.
Chapter 1 introduces, very briefly, concepts in GaN device technology that are relevant
to XRD metrology.
Chapter 5 describes in principle how measurements of peak widths can be used to give
some guide to mosaic block sizes and tilts and hence dislocation densities in GaN buffer
layers.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Section 1.2 outlines the areas in structural metrology in which XRD methods have
a key part to play and also briefly discusses future requirements and challenges
affecting structural metrology.
Osram Opto semiconductors uses Sony is using its hugely successful Playstation games
silicone encapsulants in its Golden platform to help launch the Blu-ray Disc data storage
Dragon LEDs. Shown here is the new format that uses GaN-based blue laser technology.
warm-white version.
The Korean company RFHIC says that early next year it will start
to deploy power amplifier products that feature 50 W GaN-on-
silicon transistors developed by Nitronex (inset)
Figure 1: Some examples of current exploitation of GaN-based devices in the market (images
copyright: Compound Semiconductor Magazine)
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Blue lasers based on GaN technology are expected to form an integral part of the
new ‘Blu-ray‘ electronics for high density DVD applications.
Most commonly, very thin (< 1 micron) high quality single crystal semiconductor
multi-layers, that comprise the active device, are grown by thin-film deposition
methods on appropriate substrates. Substrates are usually in the form of wafers
with diameters ranging from 1 – 30 cm and thicknesses of the order 0.5 mm. The
substrate provides mechanical stability for the device and in some cases also plays
an integral part in the device electronics.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
III IV V
4 7
C N
Carbon Nitrogen
13 14 15
Al Si P
Aluminium Silicon Phosphorous
31 32 33
Ga Ge As
Gallium Germanium Arsenic
49
In
Indium
Figure 2: A section of the periodic table showing the elements commonly used in compound
semiconductors
1.1.2 Substrates
The substrate is usually a single crystal wafer. GaN and related devices would
ideally be grown on GaN substrates. However, these are difficult and expensive to
produce. Large scale growth is currently more likely to occur, for example, on SiC
and sapphire substrates. There is a technology push towards growth on larger and
cheaper wafers using materials such as silicon.
Figure 3: Some examples of SiC (top left) and sapphire (top right) crystals and wafers that serve as
device substrates (images copyright: Compound Semiconductor Magazine)
Substrate manufacture and subsequent thin film growth can often be separate
business units.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
For substrate manufacture, large, highly perfect single crystal ingots are pulled
or cast from a melt. Thin (<1 mm) wafers are sawn from the ingots. The wafers
are mechanically and chemically polished to remove the surface layer of crystal
that has been damaged in the cutting process. They are finally etched to remove
any damage from the mechanical and chemical polishing. They are cleaned and
packaged and are then shipped to thin film deposition foundries.
layer
substrate
Figure 4: In epitaxial growth atom groups are deposited typically in monolayer by monolayer step-
flow growth.
Single crystal epitaxial semiconductor thin films may not be the only materials
types in the device architecture. Polycrystalline and amorphous solid or porous thin
films of insulating and conducting materials may also be incorporated for example
to provide insulating regions for device isolation, dielectric regions for capacitance,
or conducting regions for electrical contacts.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
GaN buffer
Substrate
Figure 5: A buffer layer of typically a few microns thick is grown on the substrate. The buffer layer
surface provides the atomic template for the subsequent multi-layer device structure.
In the case of GaN alloy layers on sapphire, SiC, or Si there is a large lattice
parameter mismatch between the crystal unit cell of the GaN alloy and the
substrate crystal. When a buffer layer of GaN is grown onto the substrate it acts as
a ‘virtual substrate‘ because it provides a growth surface which has the GaN unit
cell parameters. GaN buffer layers grown on sapphire or SiC substrates are also
called ‘relaxed layers‘ because they are not strained in order to fit to the substrate
unit cell, but have ‘relaxed‘ back to the natural bulk GaN lattice parameters
regardless of the substrate that they are on. In order for this to happen there
have to be crystallographic defects at the interface between the substrate and the
buffer layer. Some of these defects propagate through the buffer layer as it grows.
Unless steps are taken to reduce these defects at the top surface of the buffer layer,
they will also propagate into the active device region. Crystallographic defects
deleteriously affect the lifetime and performance of a device and so steps are
taken to try to reduce the defect densities at the surface of the buffer layer.
Defect reducing strategies may include the incorporation of thin strained layers
within the buffer layer, or even multiple strained layers (known as strained layer
superlattices, SLS). These are not part of the active device, but by virtue of their
strain-fields, act as a mechanical barrier to the propagation of dislocations.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Contact
p+ -AlGaN
p -AlGaN
AlInGaN MQW
n -AlGaN
Contact Contact
n+ -AlGaN
n+ -AlGaN
GaN buffer
Sapphire substrate
Figure 6: The active region of the device is grown on top of the buffer layer. It needs to be defect
free and the layer composition and thickness parameters need to be precisely controlled.
The band structure and electronic behavior is also affected by the thinness of alloy
layers. Very thin layers behave as quantum wells (QW) and have special properties
as a result.
To increase the total output from quantum wells and thin strained layers they are
often incorporated in a multi-quantum well (MQW) or multi-layer (ML) stack.
Layers of discrete islands or quantum dots (QD) can also be produced in certain
growth conditions.
The crystallographic orientation of the layers also determines how the strains and
the thinness of the layers will affect the band structure. GaN is a polar along the
[0001] direction and this also means that piezoelectric fields can be a contributing
factor in the electronic behavior of the device.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Multilayer Contact
mirror
p+ -AlGaN
p -AlGaN
MQW
AlInGaN
n -AlGaN
n+ -AlGaN Contact
n+ -AlGaN
Multilayer
mirror
GaN buffer
Sapphire substrate
Figure 7: In addition to the active region there may be photonic regions such as illustrated here
where multi-layers behave as light directing mirrors.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Figure 8: Photograph of a partially processed silicon device wafer showing the laterally repeated
pattern of copper interconnects
The dimensions of the illuminated area can range from a few tens of microns to
several millimetres depending upon the instrumentation used. Often different
areas on the same wafer are measured and the results are used to create a map of
structural properties across the wafer.
There are many processing steps after epitaxial growth. The subsequent steps
include patterning in which the area of the wafer is divided into discrete regions.
The architecture of the device then evolves in 3 dimensions. (Methods to achieve
this may involve photolithography and subsequent evaporation of dielectric,
conducting or phosphorescent layers). Eventually devices will be cleaved from the
wafer and packaged as discrete units.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
[001]
t6 GaN cap
t5 InxGa(1-x)N
t4 GaN barrier
t3 InxGa(1-x)N
t2 GaN buffer
Sapphire
counts/s
1K
100
10
0.1
0.01
0.001
16 17 18 19 20 21
Omega/2Theta (°)
Figure 9: The diffraction pattern from a layer structure contains information about the strain,
composition, thicknesses, crystalline orientation, layer quality and structural order and is like a
‘fingerprint’ of the whole device.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
GaN crystals are polar in their nature. When grown on the hexagonal basal
plane, as is common for growth on SiC and sapphire or the cubic [111] plane on
Si substrates, the resulting material has a natural polarity perpendicular to the
interface strongly influencing the electronic behavior. Irrespective of whether the
polarity is useful or not, it needs to be characterised and controlled. The issue of
how to measure the net polarity, how to grow crystals with a predictable polarity
and how polarity is distributed in domains is a hot topic for device engineers and
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
metrologists alike. Control of polarity, from the point of view of device design,
is an important issue and so growth on different orientations of substrates, for
example on different facets of sapphire is a common research theme.
There are seemingly endless possibilities for design and production of devices with
an ever increasing variation of optical and electronic characteristics. Success of a
device design depends on how well components can be alloyed and grown. The
chemistry of alloying and growth remains a challenging area and the XRD methods
remain a robust laboratory tool for crystal growers to calibrate and appraise the
outcome of their work.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
In Section 2.2 the basic concepts of biaxial elasticity as they refer to these layers
are introduced. The growth of mismatched layers causes long-range strains in
the unit cells of the epitaxial layers. Characterising these strains is important
for the prediction of electronic properties. Stoichiometric GaN, AlN and InN are
binary compounds with characteristic unit cell dimensions. By mixing quantities
of these compounds solid solution ternary phases are formed, where the unit cell
dimensions of the alloys scale in proportion to the quantities of each alloy in the
mixture. The size of the unit cell of a layer also gives a measure of the composition
of a ternary phase, but care has to be taken to treat separately the effects of
mismatch strain from compositional lattice expansion.
In Section 2.3 are introduced the properties of the crystal that disrupt the long-
range order namely the truncation of the crystal (meaning the dimensions and
shape of the crystal including layer thickness) and the presence of crystallographic
defects within the crystal. The ideal X-Ray diffraction pattern is modified by the
presence of crystallographic defects that change the observed intensities and
pattern shapes (tending to weaken and broaden them). Slight crystallographic
distortions in the vicinity of defects also add ‘diffuse‘ scatter in the diffraction
pattern.
There are different ways of expressing this periodicity. There are many different
types of crystal structure. The crystallography of GaN and related compounds as
used in epitaxial devices requires knowledge and understanding of both cubic and
hexagonal systems1.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
z
z
u
c=a c
α y
β b=a β α y
γ a
a a γ a
a
a
x x Hexagonal unit cell
Cubic unit cell
The cubic unit cell is characterised by having lattice parameters a = b = c and unit
cell angles α = β = γ = 90°.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
from the standard values will exist due to growth variations which incorporate
impurities and lattice strains. Some reference unit cell dimensions for GaN and
related compounds are listed in the table below3.
a (Å) c (Å)
Hexagonal GaN 3.1893 5.1851
Cubic GaN 4.4904 a
Hexagonal AlN 3.1130 4.9816
Hexagonal InN 3.538 5.7020
Sapphire 4.7564 12.989
SiC-4H 3.0730 10.0530
SiC-6H 3.0806 15.1173
Si 5.43105 a
z z
[uvw] [uvtw]
u
wa wc y
y
ua -ta ua
va va
x
x
Direction [uvtw] in hexagonal crystal
Direction [uvw] in cubic crystal
As for the planes, there are three Miller indices, < uvw > , to describe directions
in cubic crystals and four Miller-Bravais indices, < uvtw > , to describe directions in
hexagonal crystals, where t = −(u+v).
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
because of Bragg’s law which relates the angle of diffraction (θ) to the repeat
spacing of the crystallographic planes d(hkl). Note the use of parentheses: (hkl) is
used to specify a unique plane; {hkl} is use to specify the set of planes that share
the same atomic arrangements and have the same spacings, but are at different
orientations with respect to the crystallographic axes.
z
z
a/l
c/h
y 120o y
120o
a/k a/k
120o
a/h a/h -a/i
x x
Plane (hkl) in cubic crystal Plane (hkil) in hexagonal crystal
Figure 13: Diagrams showing how lattice planes are indexed with respect to the unit cell
There are three Miller indices, {hkl}, to describe planes in cubic crystals and four
Miller indices, {hkil}, to describe planes in hexagonal crystals where i = −(h+k).
Using three Miller indices for hexagonal systems is possible and often used, but it
does not adequately describe the six-fold symmetry of the hexagonal lattice, for
example the planes (-1-120) and (-2110) belong to the same set, but in three Miller
indices they would be written (-1-10) and (-210) respectively and could mistakenly
be thought of as belonging to different sets.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(hkl)
dhkl
(4)
(hkl)
dhkl
dh’k’l’ l ’)
’k’
(h
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
where the lines intersect the equatorial plane are represented as dots on a two-
dimensional image. They are labelled with the hkl indices of the plane.
When a new type of substrate wafer is obtained and new types of epitaxial layers
are grown, stereographic projections are often used to gain familiarity with the
orientations of the major crystallographic planes with respect to the wafer.
hkl normal
Plane normal
hkl planes
hkl
Figure 16: Diagrams showing the construction of a stereographic projection. Top left: The plane
orientation; Top right: The construction; Bottom: The projection
26
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
0002 11-22
1/d112
0001 d*(112)
11-21
000 11-20
(0002)
(11-22)
d0002 d11-22
(0001)
(11-21)
d0001 d11-21
(11-20)
d11-20
Figure 17: A 2-dimensional reciprocal lattice (top) is constructed by considering the spacings and
orientations of each set in a group of low index planes, where all of the planes lie perpendicular to
the image.
The axes for a reciprocal lattice construction also represent the real space
orientation of the crystal. For example, for epitaxial layers the reciprocal lattice is
usually constructed with the z axis normal to the surface symmetric planes and the
x and y axes approximately parallel to the interface. The reciprocal lattice directly
relates to the crystal and if a crystal’s orientation is rotated the reciprocal lattice
rotates by the same amount.
For epitaxial layers it is common to show 2-D cross sections of the reciprocal lattice
of the layers and substrates relating to a specific azimuth on the wafer surface.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(0.5,0.5,0.5)
(0,0,0)
Figure 18: In a crystal a group of atoms are associated with each lattice point. Their coordinates
relative to every reciprocal lattice point are usually given in fractions of a unit cell dimension.
Cubic GaN, whilst being metastable with respect to hexagonal GaN, is very easily
formed during growth. The lattice unit cell is face-centered cubic (fcc), there are
4 lattice points per unit cell. This is illustrated in Figure 19. The actual arrangement
of atoms in cubic GaN is shown in Figure 19. It is commonly known as the zinc
blende structure in which two atoms are associated with each lattice point of the
face-centred cubic unit cell. The positions of the atoms are given by coordinates
with respect to a lattice point. This group of two is repeated for every lattice
point and in this way the entire crystal is constructed. The figure only shows the
atoms associated with one lattice point, for clarity. Figure 19 shows the effect of
this group of atoms being repeated for every lattice point. The {111} close packed
lattice planes are illustrated. The ABCABC… characteristic stacking sequence is also
illustrated.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
[111]
CN
z
CGa
Lattice BN
point BGa
z AN
AGa
CN
y
CGa
a
x
a
a x
[-1-1-1]
z
GaN, AlN, InN (zinc blende) [111]
⊗ =
x
Figure 19: Diagrams of cubic GaN. The fcc unit cell (top left). Several unit cells filled with atoms
showing AGaANBGaBNCGaCN stacking sequence (top right). Two atoms are associated with each lattice
point in the fcc unit cell (bottom). The red sticks illustrate their tetragonal chemical bonding
arrangement (note that this is very similar to that in hexagonal GaN. The sense of bonding is
opposite (hence polar) for the directions [111] and [-1-1-1]. The coordinate set for both atoms with
respect to each lattice point (expressed as fractions of the (a,a,a) cell parameters) are as follows:
Group III atom (Ga, In, Al) (0,0,0), Group V atom (N) (0.25, 0.25, 0.25)
Figure 20 illustrates the hexagonal wurzite structure. Associated with every lattice
point is the same arrangement of atoms. For the wurzite structure four atoms
are associated with each lattice point. The positions of the atoms are given by
coordinates with respect to the lattice point. This group of four is repeated for
every lattice point and in this way the entire crystal is constructed. The figure only
shows the atoms associated with one lattice point, for clarity. Figure 20 shows the
effect of this group of atoms being repeated for every lattice point. The {0001}
close packed lattice planes are illustrated. The ABAB… characteristic stacking
sequence is also illustrated.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
BN
BGa
[0001]
z
AN
[000-1]
AGa BN
c
BGa
y
b
a AN
Lattice
x point
AGa
z
GaN, AlN, InN (wurzite)
[0001]
⊗ = [000-1]
y
Figure 20: Diagrams of hexagonal GaN. The Hexagonal unit cell (top left). Unit cell filled with atoms
showing AGaANBGaBNAGaANBGaBN stacking and illustrating polarity (top right). Four atoms are associated
with each lattice point in the hexagonal unit cell (bottom). The sticks illustrate their tetragonal
chemical bonding arrangement. The sense of bonding is opposite (hence polar) for the directions
[0001] and [000-1]. The coordinate set for all four atoms with respect to each lattice point are as
follows: Group III atom (Ga, In, or Al) (0,0,0) and (1/3, 2/3, 1/2) Group V atom (N) (0,0,0.37308) and
(1/3, 2/3, 0.89231) There is some uncertainty in the precise value of the z coordinate of the N atom
position.
Stacking faults are easily created in GaN buffer layers leading to mixtures of both
cubic and hexagonal domains and also reverse polarity domains. The control
of these domains, their relative concentrations and sizes is the subject of much
ongoing research.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
There are efforts to grow cubic GaN on, for example, (001) oriented Si substrates.
Growth of (001) cubic GaN potentially overcomes some of the polaritiy–enhanced
effects observed in (111) cubic GaN or (0001) hexagonal GaN. Also Si substrates are
a lot cheaper than the sapphire and SiC alternatives.
M-plane
A-plane
R-plane
C-plane
Figure 21: Diagram illustrating how wafers are cut from a sapphire ingot
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
y
0110 Trace of
C-plane
1210 1120
1100 1010
2429 2249
01110
11010 10110
10110 11010
01110
2249 2429
1010
Trace of
A-plane
1120 1210 (non-polar)
u Trace of Trace of
M-plane R-plane
Figure 22: Schematic stereographic projection (i.e. not to scale) of sapphire with the [0001] direction
normal to the image, showing the normals to the useful planes for XRD analysis (spots) and traces of
common wafer planes (solid grey lines).
SiC is also used as a substrate. SiC has several hexagonal forms of which SiC-4H
and SiC-6H are the most popular. Whilst the C-plane is the most common substrate
orientation, growth can be performed on wafers with other low index orientations
as for sapphire.
Si (111) wafers can be used, in particular, for AlN devices. There is increasing
interest in growth of AlGaN and even InGaN devices on AlN buffer layers on Si.
If the number of defects arising from the different thermal expansion coefficients
of the layer and substrate materials can be sufficiently reduced the use of
such a cheap substrate becomes advantageous. See Figure 23 for a schematic
stereographic projection of plane normals for a Si (111) substrate.
32
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
121
y 112
211 010
121
211
001 112 x
z
100
112 211
121
Figure 23: Schematic stereographic projection (i.e. not to scale) of cubic silicon with the (111) plane
parallel to the image, showing the normals to the useful planes for XRD analysis (spots).
33
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
be anything up to 0.2o. When there is an intended offcut the sense of the offcut, is
also indicated, usually by an angle of rotation away from the primary flat.
Normal to wafer
Normal to flat
flat
offcut
Direction of offcut
Figure 24: Diagram of a wafer (exaggerated thickness) showing the flat, the normal to the flat and
normal to the wafer (top) and the direction of offcut and the angle of offcut of the surface with
respect to the near-surface planes (bottom).
The crystal symmetry determines how many unique azimuths can be investigated
by diffraction. For example in a [0001] oriented GaN wafer similar azimuths are
each repeated with sixfold symmetry around the <0001> direction. There are two
major azimuth types, containing the plane normals to the {1120} and {10-10} planes
34
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
respectively, and one minor azimuth type containing the plane normals to the
{12‑30} planes. These are illustrated in Figure 26.
Incident beam
Normal to wafer
Diffracted beam
Flat
Azimuth
Figure 25: Diagram of a wafer (exaggerated thickness) showing the azimuth as the trace that the
incident and diffracted beam makes on the wafer surface, measured as angle phi.
Figure 26 shows a schematic representation of the plane normals for a GaN layer
illustrating the sets of planes that are commonly measured using XRD. The wafer
has a sixfold symmetry, there are 6 azimuths along which we can expect to measure
diffraction from the same types of planes.
GaN is most commonly an hexagonal crystal with the wurtzite crystal structure. It
is a polar crystal with the polar axis parallel to the <0001> direction. This polarity
can have an effect on the properties of devices. Attempts are made to avoid this by
growing more exotic epitaxial orientations for example:
{11-22} plane grown on M-plane sapphire produces semi-polar films
{11-20} A-plane GaN grown on R-plane sapphire has no polarity normal to the
surface.
35
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Trace
y of
0110
C-plane
1210 1120
1211 1121
1100 0111 1010
0110
u
Figure 26: Schematic stereographic projection (i.e. not to scale) of hexagonal GaN with the [0001]
direction normal to the image, showing the normals to the useful planes for XRD analysis (spots) and
traces of common growth surfaces (solid grey lines)
GaN can also grow in the cubic form. When grown on {0001} oriented sapphire
wafers the cubic {111} plane is parallel to the surface. The stereographic projection
of the GaN layer is similar to that for the Si{111} oriented substrate (see Figure 23).
36
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
then simplifying assumptions can be made about the strain state of the material
removing the need to make too many measurements and calculations.
y
Layer
Substrate
x
Figure 27: For analysis of epitaxial layers the axes x, y and z must be defined with respect to the
sample. z is usually normal to the layer-substrate interface. x and y are parallel to the interface.
Where possible, x, y and z are parallel to major crystallographic axes.
For the analysis the x, y, and z axes must be defined. Since the sample surface
is used for alignment of the diffractometer it is most convenient to define z as
the surface normal. x and y are therefore in the plane of the surface. Various
convenient simplifying assumptions are commonly used:
If the film has unifom thickness then the x-y plane also corresponds to the interface
plane.
If the film has uniform thickness and the surface corresponds to a crystallographic
plane then x, y and z can correspond to important crystallographic directions in the
layer
37
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
zz
zx zy
xz yz
x y
Figure 28: Stresses in the film are defined with respect to the x, y and z axes.
Unstrained or ‘fully relaxed‘: In this state the principal strains in the film are zero.
Whilst the layer unit cell may retain an orientation relationship with the substrate
the dimensions, dx , dy and dz of the layer unit cell maintain their bulk or ‘relaxed‘
values, dxr , dyr and dzr .
Fully strained or ‘coherently epitaxial‘: In this state the layer is in total registry with
the substrate across the layer/substrate interface. The layer suffers strain parallel
to the interface. The in-plane unit cell dimensions, dx and dy , of the layer are the
same as those of the substrate, dxs and dys . There is no stress perpendicular to the
interface, but the unit cell is distorted in that direction due to the Poisson effect
and so the out of plane dimension, dz is different from the layer relaxed value, dzr .
38
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
εzz
εzz
εzz εyy
εxx εyy
εyy εxx
εxx
dz
dx dx dx
dy dy dy
and (1)
The constants cijkl and sijkl are called the stiffness constants and compliances
respectively. All stress-strain calculations can be carried out using equations (1),
however, the stress and strain (second rank) tensors have each 9 components
and the stiffness and compliance (forth rank) tensors have 81 components which
39
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
together create lengthy calculations. The stress and strain tensors are symmetric
(eij = eji etc.) and the calculation can be simplified by taking account of the
symmetry using the Voigt6 notation which reduces the number of independent
strain and stress components to 6 and the number of independent compliances and
stiffness components to 36 as follows:
(2)
and
(3)
Note that the strain and stress tensors have become matrices. The factors of 2
appear in the strain matrices because the equation in (1) has reduced to σj = cjkεk.
(In the full tensor notation for example, σ11 would include two equivalent
components, c1121e21 and c1112e12 , which have now to be accounted for in one
component). The stiffness constants can be written in the form of matrices with
36 numbers comprising 21 independent terms. The general elasticity equation
becomes:
(4)
In the following text the equivalences, σ1 = σxx , σ2 = σyy , σ3 = σzz , ε1 = εxx , ε2 = εyy and
ε3 = εzz are observed.
40
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(5)
(6)
41
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
z
Z [0001] y
Y [01-10]
x
X [2-1-10]
Figure 30: Orthogonal cartesian axes X, Y and Z compared to the crystallographic axes for the
hexagonal structure: x, y, u and z
(7)
(8)
42
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Because it is assumed that σzz=0 the third equation in the matrix can be used to
solve for the strain relationship namely:
(9)
thus
(10)
(11)
Thus
(12)
where
(13)
(14)
43
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Considering the simplification for biaxial strains, as in equation (9) leads to the
relation:
(15)
(16)
(17)
Thus
(18)
where
(19)
44
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
stress-strain equations for isotropic materials are given in various forms (see basic
texts on elasticity11,12). The stress tensor is given in terms of the strain tensor by:
(20)
Where E is Young’s modulus and ν is Poisson’s ratio13, for the layer material. This
expands to give 6 independent equations:
(21)
(22)
(23)
(24)
(25)
or
(26)
45
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(27)
It is possible that the uncertainties involved in the experiment and analysis may
outweigh the necessity to calculate D precisely and that a value in the range 0.5 to
1 will suffice.
46
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
solutions can be made because for the orientations stated the principal axes used
for defining stiffness in the crystal are still coincident with the orthogonal x,y,z axes
used as the major strain axes. Other orientations require more general applications
of the elasticity equations and are not presented here.
(28)
z
[01-10]
[2-1-10] y
layer
[0001]
x
Figure 31: Orthogonal cartesian axes x, y and z defined for (01-10) oriented GaN layers
But this time introduce a general stiffness matrix with axes defined relative to the
layer orientation:
(29)
47
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(30)
(31)
C’11 is the stiffness constant along [0001] which is the same as the old C33
C’22 is the stiffness constant along [2-1-10] which is the same as the old C11
C’33 is the stiffness constant along [01-10] which is the same as the old C22 and for
the hexagonal system this is equivalent to C11
C’13 is the stiffness value relating to [0001] and [01-10] and is the same as the old C23
which is the same as the old C13 by symmetry of the hexagonal crystal
C’23 is the stiffness value relating to [2-1-10] and[01-10] and is the same as the
old C12
(32)
The equation cannot be simplified further into one single distortion coefficient, D.
(33)
(34)
48
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
z
[2-1-10]
[0-110] y
layer
[0001]
x
Figure 32: Orthogonal cartesian axes x, y and z defined for (2-1-10) oriented GaN layers
C’11 is the stiffness constant along [0001] which is the same as the old C33
C’22 is the stiffness constant along [01-10] which is the same as the old C22
C’33 is the stiffness constant along [2-1-10] which is the same as the old C11
C’13 is the stiffness value relating to [0001]^[2-1-10] and is the same as the old C13
C’23 is the stiffness value relating to [1-100]^[2-1-10] and is the same as the old C12
(35)
49
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Stress
σii
Strain εii
Figure 33: Illustration of typical stress strain curve showing linear elastic limit, elastic limit and beyond.
50
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
than 50% in going from the bulk crystal to the monolayer16. Care must be taken in
assigning high precision to calculations of composition etc. to such thin layers.
‘perpendicular‘ strain
(36)
(37)
and
(38)
The task of the XRD experiment is to measure those crystal dimensions, dx , dy and
dz as precisely as possible.
The measurement of dx , dy and dz , depends upon it being possible to find suitable
planes for measurement. A plane is suitable for measurement of strain along a
principal strain axis, if it is parallel to at least one of the other two principal strain
axes. Figure 34 illustrates this for measurement of the spacing dx that is parallel to
εxx. If the plane is parallel to both of the other principal strain axes, εyy and εzz then
the d-spacing measured is already in the sense of the principal strain axis. If the
plane is parallel to only one of the other principal axes then the correct dx value is
the projection of the d-spacing onto that axis, namely:
(39)
Where
α is the inclination of the hkl plane with respect to εxx.
51
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
zz
yy
xx
dhkl dhkl
dx dx
Figure 34 illustrates various possibilities for obtaining values dx , dy , and dz .
For planes that are parallel to only one principal axis there is the possibility of
obtaining a second d-spacing value from the same measurements. For example, in
both of the cases (1) and (2) a value for dz can also be obtained using the equation:
Note that in each case the values dxn, dyn and dzn must remain referenced to their
(hnknln) origins so that the corresponding database values for dxrn, dyrn and dzrn can
be found.
52
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
[0001]
xx
(0002)
dz
(2-1-14) d2-1-14
dx
[2-1-10] xx dy (01-15)
yy
[01-10]
Figure 35: Typically in (0001) oriented GaN, plane spacings for (2-1-14), (01-15) and (0002) are used to
measure εxx, εyy and εzz respectively.
53
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(42)
This relation is more often expressed in terms of unit cell lattice parameters, a
and c:
(43)
And
(44)
In a more general form, for an alloy AΧB(1-Χ)C made from components AC and BC:
If the composition of a layer, Χ, is unknown, then the expressions for εxx , εyy and
εzz shown in the previous section, 2.2.9 can not be solved individually, because the
X
bulk values dr etc. are not known.
To solve for Χ, the strain equations are combined with Vegard’s law and
rearranged. There are many routes through to a final solution. The complexity
of the mathematics can become high and obviously depends upon the crystal
structure of the layer and its orientation. Analytical software tools can be
created to calculate the composition for specific combinations of layer type and
measurement. As a worked example a solution for hexagonal (0001) oriented
InGaN is shown below.
54
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
InGa
InGa N
InGa
InGa
InN In0.5Ga0.5 N GaN
5.702 c
5.1851
4.9816 c
3.538 a
3.1893
3.113 a
Figure 36: In a solid solution the solute atoms are randomly sited in Ga positions (top) and have the
overall effect of expanding or contracting the lattice parameter in proportion to their concentration
(bottom).
(46)
For hexagonal symmetry it is reasonable to assume for a flat layer that εxx = εyy
hence:
(47)
55
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Substituting the strain equations (36) and (37) from above gives:
(48)
dz and dx are measured values, but dzr and dxr are unknown because the
composition is unknown. Therefore substituting the appropriate expressions for
compositions dzr and dxr namely:
(49)
and
(50)
gives
(51)
(52)
Where:
(53)
(54)
And
(55)
(56)
56
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
The distortion coefficient D has different values for different binary compounds
as shown in Table 2 in section 2.2.6. It is widely assumed, for solid solution alloys
of binary compounds, that the distortion coefficient also follows Vegard’s law.
Namely:
(57)
(58)
l
Where dxr is a unit cell dimension in the layer in its bulk relaxed state that would
be constrained to fit a parallel dimension dhx in the host surface during epitaxial
growth (see Figure 37). Mismatch is expressed numerically, e.g. as 0.0005, or as a
percentage, e.g. 0.05% or in parts per million, e.g 500 ppm.
57
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
dlxr
dhx
l
d xr d xh
Mx
d xh
Figure 37: Schematic diagram showing mismatch as the relative difference in in-plane dimensions of
the layer and substrate unit cells
The mismatch can sometimes be confused for strain, but it is not an expression of
strain. For example the mismatch between a layer and a substrate with a simple
cubic unit cell in an (001) orientation will be given by
(59)
If the layer is grown on the substrate and it is fully strained then its strain value in
the same direction is:
(60)
58
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
mismatch it is not possible to grow coherent strained layers and the GaN unit
cells grow with an orientation relationship to the sapphire substrate but are not
strained to fit the host unit cell dimensions.
59
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
dlzr
dlxr dlxs dlx
dlyr dlys dly
dsy dsx
Figure 38: The state of strain of a layer can alternatively be expressed in terms of relaxation.
Figure 29 is re-labelled here to illustrate layer relaxations of 100%, 0% and partial relaxation.
(61)
It is possible to measure relaxation >100% when the layer suffers strain beyond
that required to relieve misfit. For example if there is a non-homogeneous misfit
dislocation density or when differences in thermal expansion coefficients of
the layers and substrate give rise to internal strains on cooling from the growth
temperature.
60
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
In reality, the epitaxial thin films are not entirely perfect and this can introduce
some complexity into the measurement of d. The presence of crystallographic
defects creates microstrains in the crystal where the d-spacing is locally changed
or the crystallographic planes are locally bent (rotated). Where this kind of defect
is present in small quantities or with random orientations, it has the effect of
broadening the diffraction peaks and often reducing the diffracted peak intensity.
Such broadening can also lead to the overlapping of peaks from different layers.
Both effects can result in a reduction in the precision with which d-spacing or
orientation can be measured.
The way in which defects broaden X-ray diffraction peaks can be the subject of
further XRD studies of the microstructure of defective thin films. Descriptions of
such studies are outside of the scope of this booklet.19,20 Below are presented some
brief descriptions of defects along with a short explanation focusing only on how
they may affect the analysis of strain, composition and thickness in thin films.
61
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Figure 39: Simplistic diagrams to illustrate the effect of crystal truncation on the reciprocal lattice
spot shapes
2.3.2 Dislocations
Dislocations disrupt the long range order of the lattice. Close to the dislocations
the lattice planes are bent and the d-spacings are varied. At the core of a
dislocation the crystal planes are bent so extremely, and there are so few of them,
that they do not contribute to the Bragg peak. The total Bragg peak intensity
is therefore reduced in comparison with that from a perfect crystal. At some
distance away from the dislocation core the crystallographic planes are only slightly
distorted and so may contribute some intensity at or close to the Bragg peak. Away
from isolated dislocations the crystal is perfect and characteristic d-spacings for the
layer can be measured.
Figure 40: Diagrams illustrating dislocations. In the atomistic view looking along dislocation
lines, the dislocation core is seen as the termination of a crystal plane (left). At a lower resolution,
dislocations in an epitaxial layer are considered to have misfit components that form an array in the
layer-substrate interface and threading components that propagate through the layer to the surface
(right).
62
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Misfit dislocations are a set of dislocations that are closely spaced and located in
the interface between a substrate and a buffer layer. Their strain-fields overlap and
the overall effect of the misfit dislocation array is to accomodate the difference
in lattice spacing between the layer and the substrate. Crystal planes in the layer
above an array of misfit dislocations take on a value close to that expected for
the layer in its bulk (or relaxed) form. The misfit dislocation array is responsible
for a shift in the position of a Bragg peak between the expected position for a
fully strained layer and that for a relaxed layer. A misfit dislocation array is rarely
perfectly ordered and so whilst there is a net shift in the d-spacings of the crystal,
the corresponding measured d-spacing may be an average of a small range of
statistically distributed values21. A peak from a relaxed layer is generally weaker
and broader than that from a layer with no dislocations at the interface.
Misfit dislocations in buffer layers tend to have isolated trailing ends that thread
through the layer above the interface (so called ‘threading dislocations‘). These are
like isolated dislocations in the sense that they don’t contribute to the Bragg peak
intensity, nor to a net shift of the Bragg peak position, but rather they tend to
lower the Bragg peak intensity and contribute to broadening of the peak.
Figure 41: Mosaic blocks are considered to be the perfect regions between dislocations.
63
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Figure 42: Point defects are isolated faults in the crystal for example, impurity atoms, anti-site
defects or vacancies.
64
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Figure 43: Stacking faults are local regions where the stacking sequence is reversed.
2.3.6 Domains
Domains are regions of the crystal that exhibit subtle structural differences. For
example in GaN where polarity is an issue, regions of reverse polarity domains
may exist. It would be expected that the d-spacings of reversed polarity domains
will be the same and in that sense the presence of domains may not affect the
measurement of d-spacings and plane orientations except in the same sense as
for mosaic blocks, namely weakening and broadening of the diffraction peaks,
although the effects of this type of microstructure on strain and composition
measurements are not well known for epitaxial layers.
2.3.7 Twins
Twins are regions of material with identical crystal structure that exhibit a
crystallographic relationship with respect to each other. Their reciprocal lattices
exhibit the same orientation relationship as the real space twins. In terms of
d-spacing measurements it is likely that only one of the twinned crystals will
be measured at any time. If sets of d-spacings are to be measured it would be
necessary to make sure that all of the d-spacings in any set originate from the same
crystal.
65
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Figure 45: Twins are differently oriented regions of crystal usually connected by an internal interface
with a characteristic orientation relationship.
2.3.8 Phases
Phases are regions of different crystals that coexist within the same material. Each
phase has its own reciprocal lattice.
66
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
2.3.9 Precipitates
Precipitates are small phases of a different crystal existing within a host or ‘matrix‘
crystal. The precipitate and the host crystal each have their own reciprocal lattices.
The precipitates can form with an orientation relationship with the host or
they can form with random orientations. Peaks from precipitate phases may be
identifiable because they tend to be broader and weaker than those from the host
phase. If there are strains associated with the precipitate-host interface they may
weaken and broaden the peaks in the host material in the same way that isolated
dislocations do.
Figure 47: Precipitates are particles of one phase forming within a host matrix of a different phase.
Figure 48: Quantum dots are identical nano-sized regions of a phase. Typically they are the same
sizes and shapes. Often they are formed in regular arrays and arranged in layers.
still be available for analysis of d-spacings. It is possible, however, that for densely
packed pores the stiffness constants for the bulk material may not be applicable to
the host layer. As for quantum dots this type of analysis is still very much a research
topic for GaN compounds.
Figure 49: Pores and voids can be deliberately introduced into nano- and meso-porous materials in
order to change the material’s properties. In layered structures pores may be horizontally or vertically
aligned.
68
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
3. Principles of measurement
Some methods for the determination of the alloy composition in conventional
semiconductor materials are already well established. However, as the versatility
and scope of X-ray diffractometers increase, and our understanding of thin
films improves, new possibilities open up to improve the current methods and
calculations. It is important to note that the equipment can deliver extremely
precise measurements for even the simplest experiment. But the analyses of data
from the simplest and quickest methods can make assumptions and approximations
that reduce the precision of the final result. Some materials are not sufficiently well
behaved to lend themselves to the simplest methods. At all times it is important
not to expect the data to provide answers that are outside the scope of the
method in use.
Section 3.1 describes Bragg’s law and the measurement of Bragg diffraction peaks
in both real space angular coordinates and reciprocal space coordinates with
specific reference to how it will be applied in the reflection geometry for epitaxial
layers.
Section 3.2 describes methods for obtaining d-spacings and thicknesses from
measured data either in the form of high resolution rocking curves or high
resolution reciprocal space maps.
• Orthogonal X, Y and Z motors that drive the sample stage so that the wafer as a
whole is positioned as required with respect to the incident beam
• Phi, ϕ, or psi, ψ23 scans can be performed in order to find peaks. In an MRD,
phi rotates the sample around the stage normal and psi rotates the sample
within an arc containing the goniometer axis and the sample stage normal.
With reference to an epitaxial wafer phi is rotated to find the correct azimuth
for measurement, psi is usually used for a fine adjustment to optimise a peak
alignment.
69
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
The goniometer mechanisms in the instrument are the largest and the most
precise. They regulate and control the incident beam angle, omega, ω, with respect
to the sample stage and the detector position, 2theta, 2θ, with respect to the
incident beam direction. The incident beam angle, ω, and detected beam angle,
2θ, can be calibrated to zero positions and scanned to perform measurements (e.g.
omega rocking curves, omega/2theta scans and omega/2theta vs omega reciprocal
space maps).
y Goniometer axis
z
X-ray source
incident beam optics
x, y, z translation axes collimation and
ψ
monochromation
Sample
ϕ ω
2θ
Detector and
diffracted Stage normal
beam optics
rotation 2θ
(62)
70
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Because the angle of incidence of the beam with respect to the planes is equal to
the angle between the diffracted beam and the planes this type of scattering is
often called Bragg reflection.
Note that the apparent change in direction of the beam from the incident direction
to the scattered direction is called the scattering angle and is equal to 2θ.
The number of wavelengths, n, in the path difference is termed the ‘order‘ of
the reflection. For diffraction from crystallographic planes n is most conveniently
only considered to be 1. This is because the higher order conditions e.g. n=2, are
equivalent to first order diffraction from planes with integer spacing d/2, and
so the ‘order‘ of the reflection is accounted in the plane spacing e.g. d001, d002,
etc., rather than in the n value. Hence the more common expression for X-ray
crystallography:
n = 2dsin
(63)
Plane normal
dhkl
Path difference = n
2
Crystal planes
Figure 51: Real space illustration of the condition for Bragg reflection. Note that the incident beam
is inclined by θ with respect to crystal planes; the scattered beam is at 2θ with respect to the incident
beam and the incident beam, plane normal and diffracted beam are all coplanar.
Bragg scattering from a set of planes hkl occurs when a crystal is introduced into
the experiment such that the scattering vector Q is exactly equivalent to the
reciprocal lattice vector d*hkl . Remembering from section 2.1.1.7 that the reciprocal
lattice vector is normal to the planes hkl and has length |d*hkl | = 1/dhkl. The Bragg
condition is therefore also expressed by:
(64)
71
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(65)
In the diagram, the origin of the experiment and the origin of the reciprocal lattice
coincide. In practice this means that the center of the goniometer coincides with
the crystal (or region of the crystal) that is to be investigated. Bragg condition
is met when the endpoint of the scattering vector overlaps with the reciprocal
lattice spot. The endpoint of the scattering vector (the instrument probe) has size
and shape due to the wavelength spread in the incident beam and the angular
divergences of the optical configuration used for the experiment. The reciprocal
lattice spot has size and shape due to crystal truncation effects and lattice defects,
so the overlap of the instrument probe and the reciprocal lattice spot may not
be total at any point. However, all that is required for analyses of d-spacings
are coordinate positions of the center of the reciprocal lattice spot. The shape
of the diffraction peak namely the overlap function, is the subject of advanced
microstructural analyses and is outside of the scope of this booklet26.
Instrument probe
Q
Trigonometric relationships:
KH KO
2
|Q| = 2sin |Ko|
000
Where: |Ko| = 1/
Instrument probe
Figure 52: Top: Reciprocal space illustration of the instrument probe position in terms of the
incident and scattered wave vectors. Bottom: Combination of the instrument probe position and the
reciprocal lattice point in the condition for Bragg scattering.
72
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
2
2
Figure 53: Before a scattering measurement is performed the azimuth containing the reciprocal
lattice points of interest must be located in the same plane as the incident and scattered beams. This
is the coplanar arrangement.
73
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Qz Qz
0006 11-26
11- 0006 10-16
- 20-26
-
0005 11-25
11- 0005 10-15
- 20-25
-
0004 11-24
11- 0004 10-14
- 20-24
-
0003 11-23
11- 0003 10-13
- 20-23
-
0002 11-22
11- 0002 -
10-12 20-22
-
0001 11-21
11 21 0001 10-11 20-21
000 000 Qy
Qx
11-20 10-10 20-20
Sapphire Sapphire
11-20 flat 11-20 flat
Figure 54: Schematic cross sections through the reciprocal lattice of (0001) oriented GaN. Showing
the sets of planes in the two major crystallographic azimuths.
In the reflection geometry some reflections are not accessible because to create
the appropriate diffraction vector Q would require that either the incident or the
diffracted beam is below the sample surface. For typical wafer samples it is not
possible to obtain a measurement because the incident or scattered beam would
be totally absorbed by the sample. Such reflections are termed inaccessible.
Figures 55 and 56 illustrate some measured reflections for GaN (0001) oriented
layers on (0001) sapphire substrates for the two major types of GaN azimuth.
The table below provides approximate angles for where these reflections may be
found.
74
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
0006
0002
Sapphire
11-20 flat
Figure 55: Top: Data collected in a large area reciprocal space map showing the accessible GaN
(black) and sapphire (white) reflections found in the sapphire 11-20 and GaN 10-10 azimuth. Bottom:
Schematic diagram illustrating the trace of the diffraction plane across the GaN on (0001) sapphire
wafer from which the data were collected.
0006
10-114 -20214
0006
0002
GaN layer
Sapphire
11-20 flat
Figure 56: Top: Data collected in a large area reciprocal space map showing the accessible GaN
(black) and sapphire (white) reflections found in the sapphire 10-10 and GaN 12-10 azimuth. Bottom:
Schematic diagram illustrating the trace of the diffraction plane across the GaN on (0001) sapphire
wafer from which the data were collected.
75
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
[10-10] GaN azimuth perpendicular to sapphire [-12-10] GaN azimuth parallel to sapphire
[11-20] flat [11-20] flat
Reflection 2Theta(o) Omega(o) Reflection 2Theta(o) Omega(o)
0002 GaN 34.567 17.283 0002 GaN 34.567 17.283
0004 GaN 72.910 36.455 0004 GaN 72.910 36.455
0006 GaN 126.072 63.036 0006 GaN 126.072 63.036
10-13 GaN 63.434 (-0.323)28 -12-14 GaN 142.673 20.178
-1013 GaN “ (63.757) 1-214 GaN “ 122.494
10-14 GaN 82.051 15.881 000 6 sapphire 41.688 20.844
-1014 GaN “ 66.170
10-15 GaN 105.006 31.921 000 12 sapphire 90.738 45.369
-1015 GaN “ 73.0845
10-16 GaN 138.106 51.677 -110 8 sapphire29 61.322 9.148
-1016 GaN “ 86.429 1-10 8 sapphire “ 52.1735
20-24 GaN 109.175 11.396 -110 10 sapphire 76.899 20.948
-2024 GaN “ 97.779 1-10 10 sapphire “ 55.951
20-25 GaN 136.523 31.354 -110 14 sapphire 116.653 45.633
-2025 GaN “ 105.169 1-10 14 sapphire “ 71.020
000 6 sapphire 41.688 20.844 -220 10 sapphire 89.0348 12.279
2-20 10 sapphire “ 76.755
000 12 sapphire 90.738 45.369 -220 14 sapphire 131.18 41.339
2-20 14 sapphire “ 89.840
11-2 9 sapphire 77.2657 7.382 -330 12 sapphire 129.967 26.734
-1-12 9 sapphire “ 69.884 3-30 12 sapphire 103.233
11-2 12 sapphire 102.868 26.962 -440 8 sapphire 124.696 4.733
-1-12 12 sapphire “ 75.906 4-40 8 sapphire “ 119.963
22-4 9 sapphire 114.144 6.558
-2-24 9 sapphire “ 31.913
3.1.6 Scanning
There are many different possibilities for exploring diffraction space by scanning
either or both the instrument and sample. Both a change in 2θ, namely δ2θ, or a
change in ω, namely δω, result in a change in the diffraction vector Q, namely δQ.
A scan is simply a series of steps in δQ, where an intensity value is measured for
each step. The results for a scan are usually stored in a file and can be graphically
presented as plots of diffraction vector against intensity (in ‘diffraction space‘), or
angle (e.g. the 2theta angle) against intensity (in ‘angular space‘).
The shape and size of the recorded profile is dependent both on the size and
shape of the reciprocal lattice spot, the size and shape of the instrument probe
and the direction in which they scan across one another. In the case of d-spacing
measurements it is the position or centroid of the scanned peak that is of primary
importance.
76
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Q
Q Q
2
2
Intensity
Q
Q
Q
|Q|
Q
Q
Intensity Q
2/
Figure 57: Clockwise from top left: The diffraction vector Q is defined in terms of the wave vectors
and the incident beam angle ω and detected beam angle 2θ; a small change in either or both of the
angles results in a small change in the diffraction vector δ Q; a scan is a succession of steps in δ Q
where the intensity is synchronously measured; scans are displayed as plots of intensity against either
Q or angle.
(66)
(67)
77
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
In principle R can take on many values, for example commonly used values are:
• 2π/λ
• 1/λ
• 1
• ½
• The length of the diffraction vector |Q| is given by:
(68)
Rcos() |Qx|
Rsin()
R
|Qz|
Q
R
Rsin()
2
Rsin(2-)
R
2- Qx
Rcos(2-) Rcos()
Figure 58: Schematic diagram to illustrate the relationship between the diffraction vector and the
angles of incidence and detection
A special case of radial scan in thin films is the ‘symmetric scan‘ in which ω = θ and
the scan is then perpendicular to the sample surface. Bragg diffraction will occur if
there are crystal planes parallel to the surface and when θ = θBragg.
For other scans also known as ‘offset scans‘, ω ≠ θ and the difference between the
two values, ω - θ , is called the ‘offset‘ (o in Figure 59).
78
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Q
Q
2 2
Q
Q
Q
o
o
Q
o o
Figure 59: Top: illustration of a general 2θ/ω scan; middle: Illustration of a 2θ/θ symmetrical scan;
bottom: illustration of a 2θ/ω offset scan.
When plotted in reciprocal space units the 2-axis maps obtained can represent the
reciprocal lattice spots that they are measuring. A good representation requires
that the instrument probe size is a lot smaller than the reciprocal lattice feature
being measured. This is why for single crystal layers that have very small reciprocal
lattice spots very high resolution methods need to be used.
The figures show an example of a 2-axis scan combining a radial scan and omega
offset in high resolution. In this case the scan is performed over a small angular
range with a very small probe.
79
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
The results for a map are usually stored in a file and can be graphically presented
as 2-D plots. In ‘reciprocal lattice units‘ or ‘diffraction space units‘, the diffraction
vector coordinate (Qx ,Qz) is plotted against intensity where the intensity is
indicated for example by a color bitmap or contour scale, or quasi 3-D plot.
In angular units the scan axis positions are plotted as coordinates with, for
example, the 2theta/omega position plotted along the x axis and the omega offset
position along the y axis.
An individual scan extracted from the map can be plotted as a single scan.
(Qx ,Qz )
(2θ,ω)
A
2Theta/omega (2θo)
(2θ,ω)
2Theta/omega (2θo)
Figure 60: Clockwise from top left: A 2θ/ω versus ω map presented in angular units; A 2θ/ω versus ω
map presented in reciprocal space units; Data are extracted at offset position A in the angular map to
provide a single 2θ/ω scan.
80
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
instrument probe is very large. As it is scanned across the reciprocal lattice spot
the detail in the spot is reproduced in the scan because only one part of it is
intercepted at each omega position.
ω/2θ scan
ω
ω scan
ω/2θ scan
ω scan
ω/2θ scan
Figure 61: Top: illustration of the large instrument probe for an open detector; middle: illustration
showing the directions of an ω scan and an ω/2θ scan with the trace of the instrument probe over the
reciprocal lattice spots; bottom: the resulting ω and ω/2θ scans
81
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
3.1.7.1 Equipment
High resolution reciprocal space maps can be obtained, for example using the
X’Pert PRO MRD with an incident beam monochromator (220 Ge symmetric
2-crystal 4-reflections) and an X-ray mirror or an incident beam hybrid mirror +
monochromator combination together with a diffracted beam analyser crystal
(220 Ge symmetric 3-reflections). The sample stage is a high precision goniometer.
The X-ray mirror is not always necessary but can increase the speed of the
measurement by providing greater intensity.
Many other experimental configurations are possible and new methods may evolve
to suit specific requirements, for example, to reduce the measurement time or
increase the collection intensity.
Figure 62: Illustration and photograph of a typical experimental arrangement showing, from right to
left, X-Ray tube, mirror, monochromator, sample, analyser and detector
82
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
0 2
Cu attenuator
Z drive
Figure 63: Illustration of the experimental positions used to measure and calibrate the 2θ = 0
position
=0
Figure 64: Illustration of the experimental positions and resulting scan used to measure and calibrate
the ω = 0 position
83
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
A reflection must be chosen for measurement and ω and 2θ are set to the
appropriate values. Typically these are obtained from databases such as the
table shown in section 3.1.5. Alternatively ω and 2θ can be calculated, where θ is
obtained from Bragg’s law (see sections 3.1.2 and 2.1.1.3) and ω= (θ - φ), where
φ is the angle between the reflecting planes and the surface sample surface (see
section 2.1.1.5 for an equation for φ).
If the orientation of the wafer is known, for example with reference to a flat, then
the azimuth angle phi ϕ can be obtained from a stereographic projection or from
a database. For a symmetric reflection, ϕ can in principle take any value, but for
many of the calculations it is necessary to use the same value of ϕ that is required
for one of the asymmetric reflections. In which case the asymmetric reflection is
aligned first and the same azimuth position ϕ is maintained during alignment and
measurement of the corresponding symmetric reflection.
If the orientation of the wafer is unknown then the Bragg peak must be found
empirically, for example by setting 2θ, with an open detector, to the expected value
for a required reflection and then scanning ϕ and ω in a large 2-axis map (e.g.
ranges 360º and 10º respectively) until a peak can be found.
When a peak has been found a finer ϕ vs ω map (e.g. ranges 5º and 2º respectively)
can be performed to optimise the alignment.
84
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
this position. The substrate and layer peaks can be measured in a similar way or
both the layer and substrate can be collected in one reciprocal space map.
Section 3.2.2 describes methods for obtaining d-spacings from rocking curves by
comparison of the angular coordinates of the layer reflection position with that of
the substrate reflection.
Section 3.2.3 describes methods for obtaining layer thicknesses and multi-layer
repeat periods from fringe spacings.
85
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
If the system has been set up such that all the rotations are placed on an absolute
scale then this is equivalent to calibrating the position coordinates for the 000
lattice points to (0, 0). In this case the d-spacings can be obtained from one
reflection alone. The equations for dx and dz are simply:
(69)
and
(70)
For example, with reference to Figure 65 it can be seen that the d-spacings for
(11‑20) and (0004) in GaN can be obtained from the position coordinates of a single
11-24 reflection. This method will only provide correct results if the sample and
goniometer are precisely aligned and if there is no tilt of the GaN unit cell with
respect to the surface. i.e. ω = 0 is parallel to (0001) plane in GaN.
Qz Qx = 1/d11-20
Qz = 1/d0004
Qx
0004 11-24
0003 11-23
Q
Qz
0002 11-22
0001 11-21
2
Qx
000 11-20
Figure 65: Illustration of how dx and dz can be obtained from measurement of a single reciprocal
lattice spot when there is no layer tilt
86
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(71)
and
(72)
Measurements correspond to either the Qx-Qz plane or the Qy-Qz plane. The x, y
and z orientations are chosen to correspond conveniently to the εxx , εyy and εzz axes
respectively. Measurements from the x-z and y-z planes are treated separately. If
any averaging is required, this is done at the end of the calculation.
The values dz and dx are sometimes otherwise known as d⊥ and d||. In the case
where they correspond to unit cell dimensions, they are also sometimes termed a⊥
and a|| or a and c.
Tilts in the layer lattice with respect to the surface can arise if the substrate is offcut
and if there are dislocations at the layer/substrate interface. If the tilt is small then
the strain calculations will be unaffected. These simplified calculations rely on
the coincidence or near coincidence of the principal crystallographic axes and the
biaxial strain axes so that the assumptions of no shear stresses can be used.
If a more complete description of the strain is required because the tilt angle is
large, then an alternative analysis, employing the full strain tensor and accounting
for axial rotations, may be more appropriate.
87
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Qz
Q’x = 1/d11-20
Q0004 = 1/d0004
Qx0004
Q’x = 1/d11-20
0004
Qx11-24 11-24
Q0004 = 1/d0004
Qz0004
Qz11-24
Q11-24
Qx
000
11-20
Figure 66: Illustration of how ∆Qx and ∆Qz can be obtained from measurement of two reciprocal
lattice spots where there is layer tilt
88
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
d+d
d+d d+d
d d d
Figure 67a shows a schematic illustration of a unit cell from an epitaxial layer
which is fully strained in order to have perfect in-plane registry with the underlying
substrate. Figure 67c illustrates the unit cells of the layer and the substrate in their
fully relaxed state. Here, the lack of registry between the two in-plane lattice
parameters is accommodated by interfacial misfit dislocations. Figure 67b illustrates
an intermediate case where there is still some strain in the layer and there are
interfacial dislocations. The layer in this case is said to be partially relaxed. In all
three cases the plane spacing in the layer normal to the interface is different from
the plane spacing in the substrate and it is this difference, ∆d, that is measured.
The basic principle of the analysis is that small differences in d-spacing, ∆d, result
in changes in the Bragg scattering angle ∆θ, and also that rotations in the plane
orientation due to strain, ∆ϕ, and rigid body tilts of the layer with respect to the
interface, ∆α, rotate the Bragg peak position. All three of these effects require
rotation of the sample orientation, ∆ω, in order to collect the peak. The rocking
curve method plots intensity versus ω, and employs an open detector to catch all
possible 2θ values, so both the effects of Bragg scattering and plane rotation are
combined in the ω measurement. From a rocking curve that provides a substrate
and a layer peak, the peak separation, ∆ω, is the key value, since ∆ω=∆θ+∆ϕ+∆o.
Most of the subsequent analysis then involves identification of these separate
contributions. One of the advantages of this approach is that the open detector
maximizes the chances of capturing the peak, making the experiment more rapid
to set up. Sometimes the detector is scanned to increase the angular range of
detection, but because of the open detector, the angle 2θ is not actually measured.
89
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(73)
and for the layer:
(74)
that can be solved to give, for example:
(75)
90
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
dl = ds +∆d
θ+∆θ ω+∆θ
ds
θ ω
θ
∆θ
I
Substrate
Layer
1 2 ω
Figure 68: Illustration of how a small difference in layer and substrate d-spacing is measured as a
peak splitting in a rocking curve
(76)
If the rocking curve measurement is repeated for the azimuth, φ+180°, then the
sense of α is reversed whilst ∆θ remains the same:
(77)
and so
(78)
91
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
-
d+d d+d
d d
+ 180o
Figure 69: Illustration of how measurement of the same reflection from opposite directions is used
to measure tilt between layer and substrate
(79)
If a rocking curve is obtained for the same set of planes for the azimuth, φ+180°
then as before,
(80)
and so
(81)
92
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Furthermore, if ∆α has been obtained from a pair of symmetric rocking curves, then
∆φ can be obtained from:
(82)
(83)
(84)
therefore
(85)
The layer plane spacing perpendicular to the interface, can also be obtained from
the asymmetric measurements where:
(86)
The above analysis can be repeated for an azimuth at 90° to obtain dy.
93
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
dx
dz
d+d
s
d
s
Figure 70: Illustration of the components of d-spacing strain and tilt for an inclined plane in a
strained layer
The fine details in the diffraction patterns from highly perfect layers arise from
dynamical effects in which there are multiple interferences and reflections
throughout the crystal. Generally, for high quality epitaxial layers, details in the
diffraction patterns are best analyzed by simulation and fitting (see chapter 4)
however some simple calculations can be performed using the angular separation
of fringes to obtain approximate values for layer thickness and superlattice30
periodicity.
94
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Figures 71-73 show some examples of simulated diffraction patterns from layered
structures. Figure 71 shows the pattern around the 004 Bragg reflection from a
single 100 nm thick InGaN layer on a GaN substrate. The peaks corresponding to
the substrate and the layer can be easily identified. Around the layer peak are
fringes that are identifiable because they are equally spaced in angle, ω. Figure 72
shows the pattern around the 004 Bragg reflection from a multi-layer superlattice
with alternating 5 nm thick InGaN and 5 nm thick GaN layers repeated 50 times.
The diffraction pattern also shows thickness fringes associated with the total
thickness of the multi-layer which is 500 nm. Figure 73 shows the pattern around
the 004 Bragg reflection from a multi-layer structure containing a single layer,
multi-layer and capping layer. Features in the diffraction pattern from a more
complicated structure like this cannot be easily attributed to individual length
scales and it is more likely that the details of a measured structure like this would
be obtained through direct pattern simulation and fitting (see chapter 4).
It has been shown that for two fringes, numbered n1 and n2, that arise in the
vicinity of Bragg peaks from planes parallel to the surface32:
(87)
(88)
Hence
(89)
Where ω1 , ω2 correspond to the angular positions of the peaks n1 and n2 and
∆ω = (ω1– ω2). Note that if n1 and n2 are neighboring fringes then
(90)
And therefore
(91)
For reflections from planes that are not parallel to the interface the following
equation can be used:
(92)
95
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
counts/s
100K
10K Layer
1K Substrate
100
10 Fringes
ω1 ω2
ω2 ω1
t t
n2λ n1λ Layer
Substrate
Figure 71: Top: Illustration of layer thickness fringes in a diffraction pattern from a single 100 nm
thick InGaN layer on a GaN substrate. Bottom: The path difference between waves diffracting at
the top surface of the film and those diffracting at the lower surface is a function of the angle of
incidence and the film thickness, constructive and destructive interference effects are seen as a fringe
modulation close to the main Bragg peak.
96
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
counts/s
ω2 ω1
1K
Superlattice
100 fringes
10
0.1
0.01
Thickness fringes
35.0 35.2 35.4 35.6 35.8 36.0 36.2 36.4 36.6 36.8
Omega/2Theta (°)
ω2 ω1
Λ Λ
n2 λ n1λ
t Substrate
Multi-layer superlattice
Figure 72: Top: Illustration of layer thickness fringes and superlattice fringes in a diffraction pattern
from a multi-layer on a GaN substrate. Bottom: The changing path differences between waves
diffracting at the top and lower surface of a repeat period gives rise to fringing.
97
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
counts/s
100K
10K
1K
100
10
0.1
0.01
30 31 32 33 34 35 36 37
Omega/2Theta (°)
cap
Cap
Multi-layer superlattice
Layer
layer
Substrate
Figure 73: Top: Simulated diffraction pattern from complex multi-layer on a GaN substrate. Bottom:
The complexity of the structure makes it difficult to attribute features in the diffraction pattern to
individual length scales in the structure.
98
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
angle of incidence in air is modified to the angle of refraction within the material
and according to Snell’s law:
(93)
Where, with reference to Figure 74, r is the angle of refraction and i is the angle of
incidence. Both of these angles are measured with respect to the surface normal.
For a surface symmetric Bragg reflection, this relationship can be reworked in
respect of the real Bragg angle in the material, θm , and the observed Bragg angle
in air, θa , as:
(94)
(95)
Bragg’s law with reference to the parameters in the material is expressed as:
(96)
Replacing sinθm with an expression containing the measurable values θa gives the
equation:
(97)
This is the simplest possible correction for refractive index, more rigorous analyses
are possible. The refractive index of materials such as GaN for X-rays is typically
of the order 0.99999. The refractive index correction therefore has little impact
on the results for Bragg angles in excess of around 30° and for this reason it is
often ignored. At 30° without the correction for refractive index the d-spacing
measurements are underestimated by about 0.004%. Above this angle the error
reduces to less than 0.001% at 90o. Below 30% the error increases more rapidly
and at Bragg angle of 1° the error can be of the order 3%. The effects of refractive
index are therefore much more pronounced at glancing incidence and glancing
exit geometries and for small scattering angles.
99
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
(98)
The isotropic dielectric susceptibility, χ0, for a material can be calculated from:
(99)
i
i
a a
m m
r r
2m 2a
Material n = 0.99983
Figure 74: Illustration of the incident and refracted angles required for the refractive index
correction
100
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
The kinematic theory considers scattering from an atom or molecule in terms of its
form factor and considers how this is modified by the assembly of these units into
a crystal structure, described by the structure factor. The theory considers that the
scattering intensity from crystal units is simply added to give a Bragg peak positions
and intensities. Modifications to this are made to incorporate broadening of the
Bragg peaks from instrumental effects and crystal size effects. It is also possible
to include interference patterns from multiple layers and thin layers and methods
have been devised to model the satellite peaks from superlattices.
For highly perfect semiconductor single crystal thin films, the kinematical models
are not always adequate and more rigorous dynamical diffraction models are
more appropriate. In the dynamical models the aim is to solve the wave equation
for the entire illuminated area of the sample. The X-ray wave front is considered
to be coherent throughout the entire sample volume that is irradiated. Rather
than consider the addition of isolated scattering events the material is considered
as a medium with a varying periodic dielectric susceptibility. There is a dynamic
relationship between the exciting field of the incident beam and the excited field
of the material and together these create the internal wave-field. The dynamical
approach includes the observed effects such as extinction (reduced intensities) and
subtle peak shifts that are measurable in highly perfect single crystals.
101
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
+ +
Figure 75: Top: The kinematic theory adds contributions from different scattering regions in an
illuminated volume. Bottom: The dynamical theory considers all interactions of the incident beam
and the material in a single scattering event.
There are usually some limits on the kinds of layer structures that can be described
in the models. For example the models specify epitaxial orientation relationships
and a limited number of layer substrate relationships. Also the models need to
include descriptions of the instrument because the optics used for the experiment,
affect the resolution of the scan and the shapes and widths of the peaks. The
software may also be suitable for a number of measurement types, for example,
very high resolution omega/2theta scans or omega rocking curves. It may be
limited to particular reflections, for example avoiding extreme glancing incident or
glancing exit reflections where refraction effects become more dominant.
102
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
103
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
counts/s
1M
100K
10K
1K
100 HR scan
10
1 TA scan
0.1 Omega 17.14130 Phi 3.34 X 0.00
15 2Theta 34.54405
16 17 Psi 0.08 18 Y 0.00
19 20
Z 9.495
Omega/2Theta (°)
Omega 1.8
1.5 3.5
Single scan with small probe using a triple bounce analyser (TA) 6.9
13.7
1.0 27.1
53.5
105.6
0.5 208.6
412.0
813.9
0.0 1607.7
3175.7
6272.9
-0.5 12390.8
24475.6
48346.6
-1.0
Single scan using large probe using an open detector (HR) 95499.0
188639.0
372618.4
-1.5 736032.6
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Omega/2Theta 1453884.0
Figure 76: Top: Comparison of an HR scan with a TA scan for the same sample and reflection.
Bottom: Illustration of the different instrument probe sizes for the HR scan (large red line) and the
TA scan (small red spot).
104
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Layer:1, None, Wurtzite, thick = 7.000000 um, GaN 161389ppm, 0 0 1, Steps = 1, R% = 100.0
Layer:2.0, None, Wurtzite, thick = 0.004950 um, GaN 161389ppm, 0 0 1, Steps = 1, R% = 0.0
Layer:2.1, None, Wurtzite, thick = 0.0022um, In0.10Ga0.90N 174722ppm, 001, Steps = 1, R% = 0.0
Layer:3, None, Wurtzite, thick = 0.076900 um, GaN 161389ppm, 0 0 1, Steps = 1, R% = 0.0
Figure 77: A sample file lists all of the structural parameters for each layer in order.
4.3.3 Fitting
During fitting the simulation is compared with the measured data, then some, or
all, of the sample and convolution parameters used in the simulation model are
changed and a new comparison is made. A measure of the comparison is called
the ‘fit value‘. The fit values for a number of simulations are compared and the
set of parameters (sample and convolution parameters) whose simulation results
in the smaller fit value is selected. The process is repeated until it reaches an end
point. The selected algorithm determines the method by which the sample and
convolution parameters are varied from simulation to simulation and how an
end point is reached. The set of sample and convolution parameters that give the
smallest fit value are called the ‘best fit‘ for that particular fitting run, and can be
used to update the sample and settings files.
001822 revised 0002 ta.s00 Omega 16.93110 Phi 0.00 X 0.00
006 2Theta 33.92220 Psi 0.00 Y 0.00
counts/s
100K
10K
1K
100
10
1
0.1
15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0
Omega/2Theta (°)
Figure 78: Trial simulated diffraction patterns obtained with different parameter sets are compared
with the data until a best fit is obtained (measured data are shown in blue, simulation is shown in
red).
105
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
For example, in X’Pert Epitaxy, during fitting the following processes can be
identified:
• Smoothing the measured and simulated data and treating background intensity
• Comparing the simulation with the measured data to obtain a fit value
• Changing some or all of the simulation and convolution parameters
• Identifying a better parameter set
• Deciding when to end the fitting process and providing the best parameter set
There are many methods and algorithms available for each of these fitting steps.
Software programs offer choices from specifically adapted procedures to span a
range of fitting requirements. Some of the possible fitting algorithms used are, for
example, Smoothfit, Levenberg-Marquardt, Principal Axis, Genetic Algorithms and
Simulated Annealing.
4.4 Imperfections
Obtaining a good fit of a simulated diffraction pattern to a measured pattern can
be difficult with GaN-based device structures. The reason for this is that GaN‑based
device structures tend to contain many more defects than other compound
semiconductors that are grown on more suitable substrates.
106
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
4.4.3 Summary
Diffraction pattern simulation and fitting is used routinely for device structures
fabricated from GaN and related compounds. Because of the spreading of the
peaks and the relatively low intensities of the highest resolution scans, methods
have been developed to enhance the intensities that can be obtained from these
structures and to increase the speeds with which the data can be collected. Despite
the challenges that face simulation and fitting of diffraction patterns from GaN
and related compounds, much valuable information can be obtained from a
good, if not perfect, fit of a simulated profile to a measured profile. Certainly,
by matching the peak positions and fringe periodicities, most of the useful
information about a multi-layer structure can be obtained from the diffraction
pattern, even if the whole pattern including peak broadening is not perfectly
matched.
107
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
5. Studies of defects
5.1 Introduction
Most GaN thin films are deposited heterogeneously on various substrates e.g.
sapphire, SiC, or silicon wafers. On these substrates the lattice misfit between the
GaN and substrates is accommodated by dislocation arrays. Much of development
effort in GaN buffer layer technology is aimed at controlling the dislocation
densities and improving the crystalline quality of GaN buffer layers as virtual
substrates for subsequent device overgrowth. Strategies for improving buffer layer
crystalline quality may include patterned overgrowth, the incorporation of strained
interlayers, growing on exotic or offcut crystal orientations and composition
grading, all of which require optimization of growth conditions. There is a
requirement for measurement of crystalline quality in order to appraise the success
of any design strategy or growth condition. XRD has an important part to play
as a rapid and non-destructive method of quality assessment. This chapter briefly
describes how XRD can be employed to investigate quality issues.
The concept of mosaic blocks has been introduced in section 2.3.3. Mosaic blocks
are envisaged as regions of dislocation-free crystal. Figure 79 illustrates two mosaic
blocks in an epitaxial layer. Considering the same sets of planes in neighboring
blocks, the factors that affect the size and shape of the reciprocal lattice spots
that represent them are their dimensions and relative orientations. Crystallite size
effects are measurable, together with d-spacing variations (micro-strains) in the
direction parallel to the diffracting planes normal. Sizes parallel to the interface
are known as lateral correlation lengths and dimensions normal to the interface
109
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
are depth-wise correlation lengths. If the mosaic blocks are rotated around an axis
normal to the interfacial plane they are said to be ‘twisted‘. If they are rotated
about an axis parallel to the interfacial plane they are said to be ‘tilted‘. Some
authors use the results of XRD measurements of tilt and twist to comment on the
relative proportions of dislocation types and dislocation arrangements that are
believed to be present in the GaN layers.42,43,44
Twist
Depth-wise
correlation
length
d±δd
strain
Crystallite
size
Tilt
Lateral
correlation
length
Figure 79: Illustration of two mosaic blocks in a layer showing parameters that contribute towards a
statistical spread in the diffraction peak
110
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Finite size
and
microstrain
Surface
normal
Depthwise
correlation
Twist
length
Tilt
Lateral
correlation
length
Layer surface
Figure 80: Illustration of the orientations of spread in a reciprocal lattice spot arising from the real
space features illustrated in the previous figure
111
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
The coplanar arrangement (as previously shown in Figure 53) is the most commonly
used geometry for epitaxial layers. It offers the most precise measurements and the
instrument probe is well defined within the omega/2theta diffraction plane. For
the coplanar geometry there are the possibilities of investigating peak broadening
in symmetric reflections or asymmetric reflections.
The in-plane geometry can be used to gain access to reciprocal lattice spots from
planes that are perpendicular to the substrate-layer interface and is particularly
useful for isolating twist rotations and, in principle, for direct measurement of
lateral correlation lengths. However, due to the extended illuminated dimension of
the grazing incident beam, instrumental broadening can potentially be significant.
112
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
Coplanar
Non-coplanar
In-plane
Figure 81: Illustration of the reciprocal lattice spots that can be accessed in the coplanar, non-
coplanar and in-plane diffraction geometries
For the cases where in-plane scattering is not possible, or for a more rapid
assessment, the non-coplanar arrangement can offer an alternative geometry for
measuring the projected twist angle.
Broadening due to the lateral size effect occurs in the direction parallel to the layer
surface. For a symmetric reflection this is a direction perpendicular to the reciprocal
lattice vector and hence perpendicular to the 2theta/omega scan. For small angular
ranges it is coincident with the width of an omega scan. In reciprocal lattice
coordinates broadening due to lateral correlation lengths will be the same for all
reciprocal lattice spots (see Figure 82). Broadening due to tilt occurs tangentially
around the reciprocal lattice origin. In reciprocal lattice units, the angle subtended
at the origin is the same for all tilt-broadened reciprocal lattice spots. Therefore, by
comparing at least two symmetric reflections, for example two of the 0002, 0004
and 0006 reflections in (0001) oriented GaN, it can be established whether the peak
113
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
1/L2
δS2
1/L1
δS1
δφ2
Figure 82: Illustration of how different types of spreading can be distinguished using peak width
data from at least two coplanar symmetric reflections.
114
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
vectors are de-convoluted and for example in X’Pert Epitaxy software values for
tilts and lateral correlation length can be obtained from a single measurement
across the major axis of the ellipse. The details of the calculations are presented
elsewhere51.
δφ
000 1/L
Tilt Lateral
correlation
length
Figure 83: Illustration of how different types of spreading can be distinguished using peak width
data from one coplanar asymmetric reflection
115
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
δφ2 δφ1
000
Twist
1/L2 1/L1
000
Lateral
correlation length
δS2 δSS1
000
Finite size and
micro-strain
In-plane
Figure 84: Illustration of how different types of spreading can be distinguished using peak width
data from at least two in-plane reflections
116
XRD of gallium nitride and related compounds:
strain, composition and layer thickness
δω
δφ
000
δφ φ
Twist
Figure 85: Illustration of how different types of spreading can be distinguished using peak width
data from two non-coplanar reflections
5.5.6 Summary
At the time of writing, studies of defect structures in GaN and related compounds
are the subject of active research. Models for understanding these defects are
continuously being improved and their sophistication increases as more knowledge
is added. This chapter provides an introduction to some of the basic principles in
the analyses of defect structures. For more details on methods and calculations of
results the reader should refer to current literature52.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
References
1 For further reading see basic text on crystallography e.g. A. Kelly, G.W. Groves and P. Kidd,
Crystallography and Crystal Defects (Wiley 2000) ISBN 0-471-72043-7
2 P.F. Fewster Journal of Materials Science: Materials in Electronics 10 (1999) 175-183
3 Source X’Pert Epitaxy Materials Database
4 A. Kelly, G.W. Groves and P. Kidd, Crystallography and Crystal Defects (Wiley 2000)
ISBN 0-471-72043-7
5 J. Nye, Physical Properties of Crystals (Oxford 1995) ISBN 0-19-851165-5
6 The Voigt notation is followed here, although the Wooster notation is sometimes used, there-
fore care has to be taken when looking up numerical values.
7 W.A. Brantley, J. Appl. Phys. 44 (1973) 534-535
8 Standards on piezoelectric crystals (1949) as referenced in 5
9 J.M. Hinkley and J. Singh, Physical Review B, 42 (1990) 3546-3566
10 D.J. Dunstan, J. Materials Science: Materials in Electronics 8 (1997) 337-375
11 L.D. Landau and E.M. Lifshitz, Theory of Elasticity (Butterworth 1998) ISBN 0-7506-2633-X
12 G.E. Dieter, Mechanical Metallurgy (McGraw-Hill 1988) ISBN 0-07-100406-8
13 Poisson’s ratio is also sometimes denoted by σ (note the absence of suffixes), and the strains
more generally by uik the displacement tensor components, see ref 1.
14 A.F. Wright, J. Appl. Phys. 82 (1997) 2833-2839 and references therein
15 Values as used in X’Pert Epitaxy
16 F.D. Auret and J.H. van der Merwe, Thin Solid Films, 23, (1974) 257; Ibid., 27, (1975) 329, J. H.
van der Merwe, Philosophical Magazine A, 45 (1982) 127-143
17 Vegard’s rule is assumed for InGaN, AlGaN alloys.
18 For very small values of mismatch the difference between the two expressions can be negligi-
ble for a fully strained layer. In these cases, in particular for composition calculations, the value
for mismatch is substituted for the true strain as an approximation to simplify the mathematics
because as is a known value whereas arl is unknown.
19 See for example: P.F. Fewster, X-Ray Scattering from Semiconductors (Imperial College Press
2003) ISBN 1-86094-360-8
20 See for example: M. Birkholz, Thin Film Analysis by X-Ray Scattering (WILEY-VCH 2006)
ISBN 3-527-31052-5
21 See for example P. Kidd, P.F. Fewster and N.L. Andrew J. Phys. D: Appl. Phys. 28 (1995) A133-
A138
22 See for example, P.F. Fewster, V. Holy and D Zhi, J. Phys. D: Appl. Phys. 36 (2003) A217-A221
23 In many texts and instruments this axis is also known as Chi, χ.
24 In physics, the wave vector |K0| is more generally expressed as = n/λ. But for the purposes of
solving Bragg’s law in XRD its most convenient form is where n = 1.
25 Q is also referred to in texts as S and is also known as the ‘change of momentum‘.
26 For further discussion see for example: P.F. Fewster X-Ray Scattering from Semiconductors
(Imperial College Press 2003) ISBN 1-86094-360-8 and references therein.
27 See P.F. Fewster, X-Ray Scattering from Semiconductors (Imperial College Press 2003)
ISBN 1-86094-360-8
28 These reflections are in principle not accessible because the angles of incidence are either less
than zero or greater than 2theta. However, if there is a spread in the incident beam angle (i.e.
the instrument probe size is large) they may be partly detected.
29 The reflections marked in italics are not present in the figure because sapphire does not have
total sixfold symmetry. These reflections will be present and their opposites absent at azimuths
60° and 180° away from the one shown.
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XRD of gallium nitride and related compounds:
strain, composition and layer thickness
30 In the context of thin film epitaxy, a superlattice is a multi-layer structure in which a group of
layers are repeated many times. The superlattice exhibits a one dimensional periodicity in the
same manner as a crystallographic d-spacing but with a much larger repeat dimension, called
the superlattice period, Λ, which has values typically in the range 1 – 10 nm.
31 See for example: M. Birkholz Thin Film Analysis by X-Ray Scattering (WILEY-VCH 2006) ISBN
3-527-31052-5
32 See for example: P.F. Fewster, X-Ray Scattering from Semiconductors (Imperial College Press
2003) ISBN 1-86094-360-8; section 4.4.2.1
33 See for example: P.F. Fewster, X-Ray Scattering from Semiconductors (Imperial College Press
2003) ISBN 1-86094-360-8 ; section 2.8
34 See for example J.F. Nye Physical Properties of Crystals (OUP) 1985 ISBN:0198511655
35 See for example P.F. Fewster, N.L. Andrew, O.H. Hughes, C. Staddon, C.T. Foxon, A. Bell.
T.S. Cheng, T. Wang, S. Sakai, K. Jacobs and I. Moerman, J. Vac. Sci. Technol. B. 18 (2000) 2300-
2303.
36 See for example: P.F. Fewster, X-Ray Scattering from Semiconductors (Imperial College Press
2003) ISBN 1-86094-360-8, Takagi, Acta Cryst 15, 1311 (1962), Taupin, Bull Soc Franc Miner Cryst
87, 469 (1964), Halliwell, Lyons and Hill, J. Cryst Growth 68 523 (1984), Lyons, J. Cryst Growth
96, 339 (1989), Fewster & Curling -J Appl Phys 62, 4154 (1987).
37 V. Holy and P.F. Fewster. J. Phys. D: Appl. Phys. 36 (2003) A5-A8
38 J.F. Woitok and A. Kharchenko, Powder Diffraction, 20 (2005) 125-127, P.F. Fewster, J. Appl.
Cryst. 37 (2004) 565-574, P.F. Fewster, J. Appl. Cryst. 38 (2005) 62-68.
39 See basic text on dislocations e.g. A. Kelly, G.W. Groves and P. Kidd, Crystallography and Crystal
Defects (Wiley 2000) ISBN 0-471-72043-7
40 For a review of dislocations in GaN see e.g. S. C. Jain, M. Willander, J. Narayan, R. V. Overs-
traeten, J. Appl. Phys., 87 (2000) 965
41 H. Zhou, A. Ruhm, N-Y Jin-Phillipp, F. Phillip, M. Gross, H. Schroder, J. Mater. Res., 16 (2001) 261-
267
42 V. Srikant, J.S. Speck, D.R. Clarke, J. Appl. Phys. 82 (1997) 4286
43 T. Metzger, R. Höpler, E. Born, O. Ambacher, M. Stutzmann, R. Stömmer, M. Schuster, H. Göbel,
S. Christiansen, M. Albrecht, H.P. Strunk. Philosophical Magazine A, 77 (1998) 1013-1025
44 V.V. Ratnikov, R.N. Kyutt, T.V. Shubina, T. Paskova, B. Monemar, J. Phys. D: Appl. Phys. 34 (2001)
A30-A34
45 S. Danis, V. Holy, Z. Kristallogr. Suppl. 23 (2006) 141-146
46 V.M. Kaganer, A. Shalimov, J. Bak-Misiuk, K.H. Ploog, Phys. Status Solidi, A204 (2007) 2561-2566
47 J. Gronkowski, J. Borowski, Cryst. Res. Technol., 36 (2001) 8-10
48 See for example: M.E. Vickers, M.J. Kappers, R. Datta, C. McAleese, T.M. Smeeton,
F.D.G. Rayment, C.J. Humphreys, J. Phys. D: Appl. Phys., 38, (2005) A99-A104
49 O. Yefanov, J. Appl. Cryst. 41 (2008) 110-114
50 B. Poust, B. Heying, S. Hayashi, R. Ho, K. Matney, R. Sandhu, M. Wojtowicz, M. Goorsky, J. Phys.
D: Appl. Phys., 38 (2005) A93-A98
51 See chapter 4.7 in P.F. Fewster, X-Ray Scattering from Semiconductors (Imperial College Press
2003) ISBN 1-86094-360-8
52 See for example M.A. Moram and M.E. Vickers, Rep. Prog. Phys. 72 (2009) 036502 (40pp), and
references therein.
119