Post on 18-Jun-2020
X-ray crystallography without crystals
Iuliana Dragomir-Cernatescu: PANalyticalValeri Petkov: Central Michigan UniversityPeter Chupas: Argonne National Laboratory Thomas Proffen: Los Alamos National Laboratory
Tuesday July 31st, 2007
X-Ray Diffraction
Materials characterization via XRD
Iuliana Dragomir-Cernatescu
PANalytical Inc., USA
1) Basics of X-Ray Diffraction
2) X-Ray Diffraction Instrumentation
3) Applications to nano-materials
Outline
– Description of crystals
– Crystal structures
– Principles of diffraction
1) Basics of X-Ray Diffraction
LiquidState
Crystalline(ordered)
SolidState
Atoms, ions, molecules
GaseousState
Amorphous(disordered)
Matter
The Crystalline State
Origin: Origin: κρυσταλλοςκρυσταλλος (Greek (Greek -- ice)ice)
Crystal
A crystal is constructed by the ‘infinite’ repetition in space of identical ‘building blocks’.
Grid system
Building block
Crystal+
bb
aa
The Crystalline State
Building block describes arrangement of groups of atoms
Grid system describes how building block repeat in space
The lattice parameters describe the ‘infinite repetition’ unit. A volume element whose edges are successive grid lines.
The Crystalline State
Lattice parameters
a b c - sidesα β γ - angles
cb
a
The Crystalline State
The 14 Bravais Lattices
CubicP TetragonalPCubicI CubicF TetragonalI
MonoclinicP TriclinicMonoclinicC TrigonalR Trigonal & Hexagonal P
OrthorhombicP OrthorhombicC OrthorhombicI OrthorhombicF
Symmetry restrictions due to the lattice periodicity:
(im)proper axis of rotation: (-)1,(-)2,(-)3,(-)4,(-)6
In crystals more symmetry axis may coexist
Point groups: mathematical group of operators that leave one point fixed
There are 32 crystallographic point groups
Point Groups
32 crystallographic point groups
+
14 Bravais lattices (7 crystal classes)
⇓230 space groups
Space Groups
Diffraction is an ‘interference’ phenomenon
Waves interact with an object
Simple example: optical diffraction
Diffraction
Light of wavelength λincident on two slits ‘d’apart:
first maximum occurs when waves from each slit are exactly in phase.
i.e. when difference in path-length is exactly = x
x = d sinφ = λ
d
λ
x
φ
φ
1st maximum
λ = d sinφ
Diffraction
X-rays
λλλλλ
If we replace the ‘slit’ by an ‘atom’ and the light by
X-rays, then the atom scatters the X-rays and acts as
a point source.
Diffraction
λλ
dsinθ dsinθ
d
A C
B
B'
B"
C'
C"
A'
A"
θλ 2λ 3λ
First order Second order Third order
Diffraction
d
θ θC
D
B
nλ = 2d sinθ
B’
A
A’
The condition for all scattered waves to interfere constructively:
λ = d sinθ + d sinθ = 2d sinθ (Bragg’s law)
In a 3-d crystal the atoms are arranged in ‘planes’. The ‘incident’ and ‘scattered’ beam directions must be coplanar with the ‘normal’ to the plane (N).
Diffraction
The atoms (molecules = ‘building blocks’) of a 3D
crystal lattice lie in ‘planes’.
‘Planes’ of atoms in a crystal ≡ lattice planes
These planes are identified by the Miller indices (hkl).
Diffraction
The lattice is described by 3 axes: a, b, c.
Each ‘plane’ must intercept these axes.
The plane intercepts the axes at ¼a, ½b, c.
3Å c (???)
b
a
2Å
1Å
8Å
1Å 2Å 3Å 4Å0
c
b/k
a/h
c/l
(hkl)
(a) (b)
b
a
Diffraction
To find the Miller Indices:
– Find intercepts on a, b, c axes ¼ ½ 1
– Take reciprocals 4 2 1
⇓
– (hkl) = (421)
– All lattice planes can be indexed in the same way.
Diffraction
d100
(100)
c
b
a
(200) (110)
(110) (111) (102)
d200
Lattice Planes
Real crystal structure CsCl a = 4.11Å, λ=1.54
Calculate: d(hkl) and θhkl for the following (hkl)
hkl d θ 2θ
100
110
111
200
Lattice Planes
lkh
ad
222l) k, (h,
++=
⎟⎠⎞
⎜⎝⎛=θ2dλ
arcsin
Real crystal structure CsCl a = 4.11Å, λ = 1.54
Calculate: d(hkl) and θhkl for the following (hkl)
hkl d θ 2θ
100
110
111
200
4.11
2.91
2.37
2.06
10.798
15.343
18.935
22.006
21.596
30.686
37.870
44.012
Lattice Planes
– The reflecting power of atoms (normally called the atomic scattering factor) is related to the number of electrons in the atom.
∴ Cs + = 54 electronsCl - = 18 electrons
∴ the reflected beam from Cs+ atomshas an amplitude 3x larger thanthe beam from Cl - atoms
Wavefronts
sin θ/λ
Zf
AtomX-ray beam
Difference in phase
A
B
θ
d(100)
Cl-
Cs+
Wavefronts
strong44.01°2.055 Å200
weak37.87°2.373 Å111
strong30.69°2.91 Å110
weak21.6°4.11 Å100
I2θdhkl
To summarize:
from lattice from ‘building block’
Crystal structure
Reflection
The ‘form factor’ reduces intensities of higher angle Bragg reflections:
– Temperature factor
– Lorentz-polarization factor
– Instrumental factors
– Sample factors
Form Factor
The diffraction pattern is like a finger print of the
crystal structure:
d values reflect the unit cell parameters (‘grid’)
intensities reflect the atoms/molecules (‘building blocks’)
Summary
The Nobel Prize in Physics 1915
– What are X-Rays?
– Laboratory Systems
– Diffraction Geometries
2) X-Ray Diffraction Systems
X-rays: electromagnetic radiation with a wavelength from 0.1 Å to 100 Å (0.01 nm to about 10 nm).
What are X-rays?
Generation of X-raysContinuous radiation: caused by deceleration of electrons when passing
the positively charged nuclei in the anode or when colliding with electrons of the anode atoms.
KL
M
Radiation (Bremsstrahlung)
Decelerated electron
KL
M
Knocked-out electron
Decelerated electron
Generation of X-raysCharacteristic radiation: When an atom is bombarded with sufficiently
high energy electrons (E > Ec ) electrons can be knocked out from their shell.
Characteristic radiation: An electron from a higher shell takes the place of the knocked-out electron. The energy difference between both shells is released in the form of X-ray radiation of a specific wavelength.
KL
M
Characteristic radiation
KαKβ
Generation of X-rays
Generation of X-rays
L-shell
III
III
Kα2 Kα1
K-shell
Generation of X-rays
Kα1 and Kα2 radiation:
Kα radiation comprises two wavelengths: Kα1 and Kα2.
The wavelengths correspond to the transitions from the L-shell to the K-shell. The L-shell has three energy levels from which level I is empty.
Generation of X-rays
• Electrons are emitted by a hot filament
• High voltage accelerates electrons
• Electrons bombard anode material at high speed
• Kinetic energy of electrons largely transferred into heat and X-ray radiation
Current (mA)
Voltage (kV)
Spectrum of X-rays depending on applied voltage and anode material. Characteristic radiation is used for experiments.
Mo-anode
Continuous radiation
Characteristic radiation
Generation of X-rays
X-r
ay In
ten
sity
(rel
ativ
e u
nit
s)
Wavelength (Å)
3
1.0 2.0 3.00
1
2
0
4
5
6
SWL 510
15
20
25kV
Kα
Kβ
XRD uses a small partof the X-ray spectrum
What are X-rays?
Rontgen – Nobel Prize in Physics 1901
Debye-Scherrer X-Ray cameras – Circa 1920
Norelco (Philips) XRD Serial #2 Circa1942
X-ray tube
Soller slit Soller slit
Anti-scatter slit
Receiving slit
Monochr.
Divergence slit
Sample stage
Mask
Detector
GoniometerModern Powder Diffractometer
High Resolution Diffractometer
High Resolution Diffraction - Materials Research Diffractometer
monochromator Symm. Ge[220] 4 - Crystal
or Asymm.
(Perfect) epitaxial layer,stressed and textured sampleshighly textured layers
X-ray tube(line focus)
Soller slits(optional)
X-ray mirror
Divergence slit
Detector 2
Triple AxisSectionDetector 1
Optical slit
+ all kinds of applications+ all kinds of applications+ interchangeable optics+ interchangeable optics
Bragg Brentano Para-Focusing Diffractometer
• Sample surface bisects incident and scattered beams
• Scattered beams focus at the same distance as the tube focus in receiving slit
• Optimal resolution
λ = 2 d sin(θ)
θ θ
Goniometer circle
Specimen
Focusing circle
tube focus
Soller slit
divergence slit anti-scatter slit
receiving slit
Soller slit
diffracted beam
monochromator
detector
width mask
Classical Powder Diffractometer
Optical components in the X-ray beam path
Incident Beam Monochromator
X-ray tube(line focus)
Incident beam monochromator
Irradiation slit
Programmabledivergence slit
Soller slits
Detector
Polycrystalline sample
Anti scatter slit
Receiving slit
Soller slits
line focus X-ray tube
Curved Ge(111) incidentbeam monochromator
divergence slit
Kα1
Kα2
Johansson Monochromator
The symmetrically cut curved Ge(111) monochromator in combination with the divergence slit filters out the Kα2component leaving a beam with Kα1 radiation only.
X-ray tube(line focus)
Samples with unevensurfaces
Divergence slits Soller slits
X-ray mirror
Soller slits
Divergence slits
X-ray mirror
Receiving slit
Detector
The Parallel Beam Geometry
The Capillary Spinner
Powder sample in capillary spinner
X-ray tube(line focus)
Divergence slit
Hybrid monochromator
Δ θ = 18" - 25"
Soller slits
Anti-scatterhousing
Soller slit
X’Celerator
Anti-scatter shield
Anti-scatter shieldfor capillary spinner
High Resolution Diffractometer
ω
χφ
detector
sample
ω2ω’ X-rays
ω
χφ
detector
sample
ω2ω’ X-rays
monochromator Symm. Ge[220] 4 - Crystal
or Asymm.
(Perfect) epitaxial layer,stressed and textured sampleshighly textured layers
X-ray tube(line focus)
Soller slits(optional)
X-ray mirror
Divergence slit
Detector 2
Triple AxisSectionDetector 1
Optical slit
3) Applications Examples
• Phase ID, quantification• Crystal Structure• Micro-structure (crystallite size and non-uniform
strain)• Residual stress• Texture• Advanced characterization of layered structures
(LEDs, High-frequency IC’s, IR optopelectronic, Thin film recording media, reading heads, etc.)
The Reciprocal Lattice – design your experiment
Create reciprocal lattice (RL), where each point represents a set of planes (hkl)-The points are generated from the RL origin where the vector, d*(hkl), from the origin to the RLP has the direction of the plane normal and length given by the reciprocal of the plane spacing.
000
001
002
110
111
112
d*(11
2)
1/d112
001
002112
111110
1) Bragg’s law concept is a simplification that is useful in a limited number of situations
2) XRD methods are advancing, we need a clear way of understanding them all.
Reciprocal Lattice of a Single Crystal in 3D
004
113
224
115
440-440
d* | d*| = 1/dhkl
Just a few points are shown for clarity
•There are families of planes
•All planes in the same family have the same length |d*|, but different directions
•The family members have the same 3 indices (in different orders e.g. 400,040,004 etc)
-2-24
Reciprocal Lattice of Powder vs. Single Crystal
004
113
115
400
d*
PowderSingle Crystal
2D view
powder textured single crystal
05-05-2005
Symmetric “powder” scans2Theta/Omega scan
scattering vector S
05-05-2005
Symmetric “powder” scans2Theta/Omega scan
111
05-05-2005
Symmetric “powder” scans2Theta/Omega scan
111
220
311
05-05-2005
Symmetric “powder” scans2Theta/Omega scan
111
220
311
004 331
05-05-2005
Symmetric “powder” scans2Theta/Omega scan
111
220
311
004 331422
511
Conversion of silica diatoms to MgO diatoms
• Diatoms are single-celled aquatic micro-organisms that assemble complex silica (SiO2) micro-shells (frustules) containing channels, pores, protuberances, or other fine features arranged in intricate patterns.
2Mg + SiO2 => 2MgO + {Si}
shape-preserving conversion
T=700oC
SiO2 diatom (aulacoseira, sp.) Converted MgO diatom
Gas/Solid Displacement Reaction:
1μm
Phase Quantification
X-ray diffraction pattern of the sample annealed for 45 minutes.
20 40 60 80
0
2000
4000
6000
8000
10000
MgO
Mg 2S
i
Mg 2S
i
MgO
Mg 2S
i
Mg 2S
iM
gOM
g 2Si
SiO
2Mg 2S
i
Mg 2S
i
Mg 2S
iMg 2S
iMgO
Mg 2S
iM
gOS
iO2M
g 2Si
Mg 2S
iS
iO2In
tens
ity [c
ount
s]
2θ [degrees]
Phase Quantification
0 50 100 150 200 250 3000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Wei
ght F
ract
ion
M g2Si; M gO ; S iO 2
Tim e [m inutes]
The weight fraction of the MgO, SiO2 and Mg2Si as a function of the reaction time.
Micro-structure from XRD pattern
30 40 50 60 70 80 90 100 110 1202Theta (°)
2
4
6
8
Inte
nsity
(cps
)
5 nm crystallites35 nm crystallites
44 45 46 47 48 49 50 512Theta (°)
1
2
3
4
Inte
nsity
(cps
)
The periodicity of the crystal lattice ends at the crystallite boundaries.
Crystallite size effect – spherical crystallites• The diffraction lines become broad when the crystallites are small
• The FWHM of a given hkl diffraction line is inverse proportional with the crystallite dimension in the hkl direction
• In the case of spherical crystallites for every single hkl the FWHM is the same
FWH
Md*
Micro-strain
• Micro-strain is a non uniform strain in the crystalline lattice created by defects: dislocations, precipitates, stacking faults, etc.
Micro-strain and crystallite size effect in RS
• Unlike the crystallite size effect, the micro-strain effect becomes more enhanced at higher d* values
• This give the possibility to separate the two effects
FWH
M
d*
Crystallite size and micro-strain during diatom conversion
• The evaluation was performed using the Warren-Averbachmethod
• Crystallite size and micro-strain was monitored as a function of annealing time
• The micro-strain decreases as the annealing time increases, where the crystallites become larger as the time of annealing increases
Median of the size distribution and micro-strain as a function of annealing time.
0 60 120 180 2400.0
0.1
0.2
0.3
0.4
(<ε2 >)
1/2 [%
]
Time [minutes]
5
10
15
20
25
30
med
ian
[nm
]
Titania nanotubes – Phase IDHigh surface area photocatalytic nanotubes
– anatase phase more desirable
Cellulose whiskers used as template
Titania nanotubes - quantification
Calcination removes cellulose and produces nano-tubes with various size crystallites
TEM images of (a) hollow titania nanotubes derived from 10 TALH/PDADMAC bilayers calcined at 600 ° C and (c) high resolution TEM image
-0.50 -0.25 0.00 0.25 0.50
0.0
0.2
0.4
0.6
0.8
1.0 calcinated at 525oC D = 7.5 nm
calcinated at 600oC D = 20 nm
Nor
mal
ized
Inte
nsity
d* [1/nm]
XRD analysis of old paint layers
Armida is watching the destruction of her palace,
Ch.A. Coypel (1694 -1752)*
Sample 2: blue drapery of Armida* Sample courtesy of Academy of Fine Arts, Prague, Czech Republic
Position [°2Theta]30 40 50 60
Counts
0
1000
2000
Spot 1
Compound Name Chemical FormulaCerussite, syn Pb C O3Quartz $GA, syn Si O2Hydrocerussite Pb2 O C O3 ( H2 O )2Hematite, syn Fe2 O3
Position [°2Theta]30 40 50 60
Counts
0
2000
4000
6000
Spot 2Compound Name Chemical FormulaCerussite Pb C O3Hydrocerussite Pb3 ( C O3 )2 ( O H )2Lazurite Na8.16(Al6 Si6 O24)(S O4)1.14 S.86Cristobalite Si O2Quartz, syn Si O2
05-05-2005
Grazing Incidence diffraction geometryGIXRD 2Theta scan
Glancing IncidenceGIXRD 2Theta scan
Glancing IncidenceGIXRD 2Theta scan
Glancing IncidenceGIXRD 2Theta scan
Glancing IncidenceGIXRD 2Theta scan
Phase ID – Depth profiling
Cu(In,Ga)Se2 solar cells
GIXRD - Thin film phase analysis
X-ray tube(line focus)
Soller slits
X-ray mirror
Thin layers
Detector
Sample
Parallel platecollimator
Incident angles
ZnO
ZnO
CuGaInSe
CdSe/Mo
ZnOCuGaInSe
Effect of dislocations on the XRD pattern
• Similar with TEM experiments the effect of dislocations is not visible when gb = 0
0 2 4 6 8 10 120.00
0.01
0.02
0.03
0.04
0.05
0.06nano-Cu deformed under liquid nitrogen74% reduction 331
222
311
220
200
111
β [1
/nm
]d* [1/nm]
0≠⋅bg
0≅⋅ bg
Dislocations character and density in bulk nano Cu
• Nanostructured Cu was obtained through severe plastic deformation (in the present case by rolling under liquid nitrogen)
• Dislocation density and character was determined from the XRD pattern
65 70 75 80 85 90 95 100
0
20
40
60
80
100
rolling reduction [%]
Dis
loca
tions
Cha
ract
er [%
]
0.5
1.0
1.5
2.0
edge
screw
ρ [x
1015
m-2]
ρ
0 50 100 150 2000.00
0.01
0.02
0.03
0.04
0.05
67% reduction
74% reduction
84% reduction
97% reduction
crystallite size [nm]C
ryst
allit
e S
ize
Dis
tribu
tion
Func
tion
1010
2110
ELOG region
4μm 6μm
Sapphire substrate
SiO2 SiO2 SiO2GaN buffer layer
Sample courtesy of D. Cherns and S. Henley, H.H. Wills Laboratory, University of Bristol, UK
Cross section Plan View
Epitaxial Lateral Over Growth (ELOG)
ELOG region
• GaN on Sapphire with SiO2 strips• Omega scan shows in-plane orientation• RSM shows different in-plane spacing between GaN buffer layer and GaN lateral overgrown layer
Dislocations in GaN devices
Texture– non random orientation of crystallites
S = 1/dhkl
Spherical shell radius 1/dhkl
2θ
2θS
1/dhk
l
ω
χφ
detector
sample
ω2ω’ X-rays
Sampling the intensity distribution over a given hkl shell.
ZnO nano-belts - Texture Analysis
ψφ ψ = 610
φ
ψ
ψ = 610
φ
ψ
• Showing the orientation of the ZnO nano-belts:– 6 crystallographic orientations
SEM image 0002 pole figure Schematic representation of the 6 orientations
Correlation between pole figures – 6 crystallographic orientations
0001 10-10
11-2011-22
10-11
Orientation of ZnO with respect to Al2O3 substrate
ZnO 0001Al2O3 110Al2O3 001Al2O3 100
ZnO 10-10 ZnO 11-20
ZnO 10-11
001
0001
10-10
100
10-11
Large lattice mismatch prohibits the formation of large area epitaxial relations.
Micro-Diffraction
X-ray tube(point focus)
Δθ= 0.3°
Sample with small areaof interest Mono-cap
X’Celerator
0.4 0.4 mm
Cu plating
Cu(111)
xx
In-plane Diffraction
XX--ray lensray lens
CrossedCrossed--slits slits 0.1 x 5 mm0.1 x 5 mm2222θθ
Parallel plate collimatorParallel plate collimator
ωω, , φφ
Co
CrNiPAl
Textured polycrystalline Co(CrPtTa) alloy layers in hard discs
Co-based magnetic thin film• typically 25nm thick•hexagonal phase•highly textured
Polycrystalline Cr
Polycrystalline textured Al
Amorphous NiP
Optics: X-Ray lens, Soller slits, Parallel plate collimator
Al (200)
Co (002)
Co (100) Co (101)
Amorphous NiP
40 45 50°2Theta
0
200
400
600
800
counts/s
Co(
100)
Co(002)
Co(
101)
In-plane diffraction geometry
Co(100)
Co(002)
Co(101)
Conventional diffraction
Conventional diffraction geometry vs. In-plane
Thin Film Characterization by X-rays •• Pseudomorphic epitaxial layers. “No” defects. Strain may be present
Example : AlGaAs/GaAs, SiGe/SiApplications: Lasers, High-frequency IC’s
• Lattice mismatched epitaxial layers. Layers are partly (or fully) relaxedExample: Strained Si, ZnSe/GaAs, InAsSb/GaSbApplications: Blue LED’s, IR optopelectronic
• Layers with large lattice mismatch and/or dissimilar crystal structuresExample: GaN/Sapphire, YBaCuO/SrTiO3, BST, PZTApplications: Blue Lasers and LED’s, High Tc Superconductors,
Ferroelectrics• Layers where the epitaxial relationship is weak. Highly textured.
Example: AuCo multilayers on SiApplications: Thin film media, heads
High Resolution Diffraction - Information from RS
•• shapeshape
kkii
ωω
kkhh
22θθ
•• positionposition
symmetricasymmetric
In-plane
Range of tiltsSpread due to finite size effects layer thickness
Tilt, Thickness and Lateral Width
Strained Layer
004 224
002
006
-2-24
SubstrateSubstrateLayerLayer
Q||220110
Q⊥
aS
fully strained
at=aS
S
L
The in-plane lattice parameter of a fully strained layer matches that of the substrate.
Relaxed Layer
004 224
002
006
-2-24
fully relaxed
aL
at= aL
L
S
Δ at
SiGe devices characterization
-4000 -3000 -2000 -1000 0 1000 2000 3000Omega/2Theta (s)
0.1
1
10
100
1K
10K
100K
1M
10Mcounts/s
Si substrate
SiGe
SiGeSi cap
Ge %
12.7
0
2
4
6
8
10
12
14
0 20 40 60 80 100
Thickness (nm)
Ge
(%)
Si
SiGe
Reciprocal Space Mapping
-100 -50 0 50 100Qx*10000(rlu)
5600
5620
5640
5660
5680
5700
5720
5740
Qy*10000(rlu) #1_M1.A00
1.6
3.0
5.4
9.8
17.9
32.5
59.0
107.3
195.0
354.5
644.5
1171.6
2129.6
3871.2
7037.1
12792.0
23253.1
42269.2
76836.5
139672.5
253895.1
Graded SiGe to 20%(relaxed)Si0.8Ge0.2
Si substrate
Strained SiSiGe 5x
Si(004)
SL
SiGe
Ge gradient
Al0.3Ga0.7As 0.02μm
T
Period?
Size?Relaxation?Strain distribution ?Composition ?
Periodic Spacing?
Al0.3Ga0.7As 0.02μm
GaAs
GaAs 0.02μm
20x{GaAs+InAs}
Vertical correlation?
InAs/GaAs Quantum Dot Structures
XRD results were compared with EDX+TEM analyses
(Fewster ICMAT 2001)
Composition and sizeanalysis of QDs usingIn-plane scattering onan MRD.
Simulation and modelling:
Quantum dot analysis
Buffer Layer StructuresRelaxed Buffer layers as virtual substrates:e.g. Si/Ge on Si
InGaAs on GaAsGaN on Sapphire
Substrate and surface layer lattice parameter calculations from reciprocal lattice coordinates (Bragg’s Law)
Graded InxGa(1-x)As Buffer layer with dislocations
GaAs substrate
InP capping layer
d*substrate
d*layerd*cap
tilt
P. Kidd et al, J. Crystal growth, (1996) 169 649-659
Bent multilayer sample
4.8o
InGaAs tensile and compressive alternating multilayer on 001 InP substrate.
Samples with Bend or Tilt
DHS 900 Domed Hot Stage
X- ray tubePrimary optics
X- ray mirror
X’Celerator
X- ray tubeLine focus
Ge [220] 4- crystalmonochromator
Beam size:1.4 x 10 mm2
Fast X-ray set-up with X’CeleratorX-ray diffractometer used
High incidence (11–24) scattering geometry
X’Celerator
In-situ high-resolution diffraction studies on the thermal stabilityof MOCVD grown InGaN/GaN
Thermal stability studies
0
100
200
300
400
500
600
700
800
900
0 100 200 300 400 500 600 700
time (minutes)
T (d
egre
es)
-39800 -39600 -39400 -39200 -39000Qx*10000 (rlu)
47800
48000
48200
48400
48600
48800
49000
49200
49400
Qz*10000 (rlu) 27°C.y00
1.5
2.4
4.0
6.6
10.9
18.0
29.7
49.0
80.7
133.1
219.3
361.4
595.7
981.7
1618.0
2666.6
4394.7
7242.9
11937.0
19673.3
32423.4
SL0
GaN
SL+1
-39600 -39400 -39200 -39000 -38800 -38600Qx*10000 (rlu)
47600
47800
48000
48200
48400
48600
48800
49000
49200
Qz*10000 (rlu) 800°C.y00
1.4
2.3
3.7
6.0
9.7
15.5
25.0
40.2
64.6
103.9
167.0
268.5
431.8
694.2
1116.1
1794.4
2885.1
4638.6
7458.0
11990.9
19278.9
GaN
20 minutes each RSM
Summary
Present phases, crystalline structure, quantification, texture, macro and micro-stress, crystallite size, shape, distribution, defects density, defects type, thin film thickness, composition, mozaicity, mismatch, etc.
X-ray crystallography without “usual”crystals: essentials
Valeri PetkovDepartment of Physics, Central Michigan University,
Mt. Pleasant, MI 48859petkov@phy.cmich.edu
What is a “usual” crystal ?
Atoms in crystals sit on the vertices of 3D periodic lattices…
Diamond
“Crystal Structure” = Lattice type and symmetryUnit cell parameters:
a, b, c, α, β, γAtomic positions inside the unit cell: (x,y,z)…..
Allow to compute and predict properties of crystals..
PHY 101: Diffraction of light
First Nobel Prize to Rntgen (1901) – x-rays discovered !!
X-rays in use: 1914/1915 Nobel Prize – Laue/Bragg
XRD crystal structure determination
Theory:
Structure of “usual” crystals: A success story….
KBr (Laue) Icosahedral Quasicrystal Protein Crystal
Nobel Prize (2006), again….
Powder XRD: also a successful story
Single crystal
Powder of many crystallites
Many materials are not large i.e. “single” crystals,rather they come as a collection of many small ( ~ :m) crystallites - “powder” crystals
Traditional crystallography stillworks pretty well: Rietveld-type analysis…..
However, a great deal of materials are “not-usual” crystals.
Examples:
• Bulk crystals with substantial intrinsic disorder – (In/Ga)As• Very small (nanosize) pieces of usual crystals – Cd(Te/Se) quantum
dots• New materials – V2O5 nanotubes
“Usual” X-ray (Bragg) diffraction is difficult to apply in such cases..
Why ?
What can we do to: Determine the “3D structure” ? Find the average “crystallite/domain” size and “lattice” strain ?Perform “phase” identification ?
“Usual” Crystals
Bragg peaks only Both Bragg peaks and diffuse scattering
5 10 15 20 25
0200
400600800
1000
120014001600
18002000
Inte
nsity
(a.u
.)
Bragg angle, 2θ
“Unusual” crystals
Diffraction patterns of “usual crystals” show many well-defined Bragg peaks. Diffraction patterns of “unusual” crystals show both Bragg-like peaks (not so many, not so sharp) and diffuse scattering (that may not be neglected). Diffraction patterns of “non-crystals” (glasses, polymers, liquids) show diffuse scattering only.
Simulated
2d patterns
1d patterns
Glasses, liquidsLong-range (~mm), periodic order Limited (~ nm) but measurable order Short-range ( sub-nano) order only
Diffuse scattering only
Simulated
2d patterns
1d patterns
0 5 10 15 20 25 30 35 400
5
10
15
20
25
Inte
nsity
, a.u
.
Q (Å-1)
5 10 15 20 25
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Inte
nsity
(a.u
.)
Bragg angle, 2θ
Diffraction patterns from materials with different degrees of structural coherence/size/periodicity
So, what can we do ? Total XRD and Atomic Pair Distribution Function Analysis (PDF)
Q=4πsin(θ)/λ=1.0135sin(θ)E[keV]
S(Q)=1+ [ ] [ ]22. )(/)()( QfcQfcQI iiiiel ∑∑−
G(r) = (2/π) ∫=
−max
,)sin(]1)([Q
oQ
dQQrQSQ
G(r) = 4πr[ρ(r) - ρo]ρ(r) is the local andρo the average atomic density
Diffraction experiment
5 1 0 1 5 2 0 2 5
02 0 0
4 0 06 0 08 0 0
1 0 0 0
1 2 0 01 4 0 01 6 0 0
1 8 0 02 0 0 0
Inte
nsity
(a.u
.)
B ra g g a n g l e , 2 θ
The atomic PDF peaks at characteristic interatomic distances reflecting the 3D structure of materials. Total scattering atomic PDF: 1D map of all interatomic distances, no long-range order or periodicity implied. i) Need x-rays of higher energy – to reach higher Q
ii) Need stronger flux & more efficient detectors – to measure the diffuse component of XRD pattern
Instrumentation ?
Total XRD Instrumentation: in-house
In-house set upE (x-rays) ~ MoKa 17 keV/8=0.71 A
Ag Ka 22 keV/8=0.55 AQmax ~ 16-20 A-1
CMU, Department of Physics
Mo
In-hose data/X’Pert diffractometer: Si standard
0 20 40 60 80 100 120
0
15000
30000
45000
60000
75000
90000
0 2 4 6 8 10 12 14 16
0
2
4
Stru
ctur
e fu
nctio
n Q
[S(Q
)-1]
Wave vector Q[A-1]
(....)(400)
(222)
(311)
(220)
(111) Si standard Mo Ka, X'Pert
Inte
nsity
Bragg angle, 2 theta
(....)
(222)
(400)
(311)
(220)
(111)
SiS.G: F d 3 m (227) Structure: diamond typeCell parameters:a=b=c=5.4309 A
α= β= γ= 90.0° Si (8a) 0.125, …
Reciprocal/diffraction space Real space
0 10 20 30 40 50
-0.4
-0.2
0.0
0.2
0.4
0.6
0 2 4 6 8 10
-0.4
-0.2
0.0
0.2
0.4
0.6
G(r)
Radial distance [A-]
G(r)
Radial distance r [A]
(....)
(12)
(6)
(12)(12)
CN= (4)
Fouriercouple
Total XRD Instrumentation: synchrotron
Synchrotron x-rays Continuum of wavelengths Energy range (0 ~ 150 keV vs 8 keV from Cu tube)
(Advanced Photon Source, Argonne, Chicago)
390 meters (1,225 feet)
Inside the hutch (APS):
MAR345; GE: exposure ~ sec Ge SSD ~ 105 sec (10 h)
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00
5
1 0
1 5
2 0
2 5
Inte
nsity
, a.u
.
Q ( Å - 1 )
Peter Chupas will give more details !
Synchrotron x-rays: 100 keV/ 8 ~ 0.1 A; Qmax ~ 40-50 A-1
What about using neutrons ?
Thomas Proffenwill tell you more..
Zinc-blende type structure:a(GaAs)=5.653 Å;a(InAs)=6.038 Å
- In,Ga (0,0,0)
- As (1/4,1/4,1/4)
Vegard’s law holds:
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.010
20
30
40
50
60
70
GaP
GaAs
Maycock, Solid State Electronics 10 (1967) 161.
Ther
mal
con
duct
ivity
(W/m
.K)
Composition (x in GaAs1-xPx)
Properties show nonlineardependence on concentration, x.
5 .5
5 .6
5 .7
5 .8
5 .9
6 .0
6 .1
Latti
ce p
aram
eter
However,
Crystals with substantial intrinsic disorder: (In/Ga)As semiconductors
What is going on ?
Crystals with intrinsic disorder: (In/Ga)As semiconductors
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 50
2 0
4 0
6 0
8 0
1 0 0
1 2 0
1 4 0
Q ( Å - 1 )
W a v e v e c t o r Q ( Å - 1 )
Inte
nsity
, a.u
.
I n 0 . 3 3 G a 0 . 6 7 A s
2 0 2 5 3 0 3 5 4 0 4 5
0 . 0
0 . 5
1 . 0
1 . 5 H i g h - Q
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5
- 2
0
2
4
6
8
I n 0 . 3 3 G a 0 . 6 7 A s
Q[S
(Q)-
1]
W a v e v e c t o r Q ( Å - 1 )
Very little structure/peaks is evident in the raw data at high-Q (inset to top panel). However, significant oscillations (i.e. information) is present in the total XRD/structure function S(Q) extracted from the raw XRD data.
It becomes evident by dividing the raw data to <f(Q)>2 and multiplying by Q.
The structural information is there;we just have to measure & take it into account.
In0.5Ga0.5As
In0.5Ga0.5As
With E = 60 keV Qmax= 45 Å-1
With E= 8 keV (I.e. Cu Kα) Qmax = 8 Å-1 only
Crystals with intrinsic disorder: (In/Ga)As semiconductors
E=60 keV
FourierTransform
Significant Bragg scattering is present in the S(Q)s of the end members GaAs and InAs. The materials are perfectly crystalline. The Bragg peaks disappear at much lower Q
values in the alloys. At high Q values, only oscillating diffuse scattering is present. The
alloys exhibit significant local positional disorder due to the presence of two distinct
bond lengths -Ga-As and In-As. These bonds are seen as a split first peak in the
experimental PDFs.
Petkov et al. PRL 83 (1999) 4089
Mo/Ag Ka
Do we really need total XRD/high-Q data ?
Real-space resolution = 2B/Qmax ~ 0.15 Å with Qmax = 45 A-1
The experimental PDFs of the alloys can be fit
only if both As and metal (In,Ga) atoms are allowed to be statically displaced
from their positions in the ideal zinc-blende
lattice.
The experimental PDFs of the end members can be fit well with a structure model
based on the perfect zinc-blende lattice.
In,Ga
As
Schematics of the discrete atomic displacements in In-Ga-As alloys. The ideal lattice(thin line) can be
compared with the distorted lattice (thick line).Exp. Data - symbolsFits - red line
As
Ga,In
ab
c
Crystals with intrinsic disorder: (In/Ga)As semiconductors
Here is how the zinc-blende lattice distorts locally to accommodate the two distinct Ga-As and In-As bonds present of In-Ga-As alloys.
Both As and metal (In,Ga) atoms are displaced from their positions in the
ideal zinc-blende lattice.
The rms deviations (effective thermal factors) of both As andmetal (In,Ga) atoms increase.
The lattice distortions/strain are more pronounced on the As thanon the metal (In,Ga) sites.Results from the crystal structure refinements
based on the experimental PDFs
0.002
0.004
0.006
0.008 (b)
Latti
ce p
aram
eter
(Å)
Dis
cret
e di
spla
cem
ents
(Å)
(Ga;In)
As
<u2 >
(Å2 )
Composition (x in InxGa1-xAs)0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12(c)
As
(Ga;In)
5.55.65.75.85.96.06.1
(a)
The average structure preserves its cubic symmetry
Petkov et a. Physica B 305 (2001) 83.
New Materials: V2O5 nanotubes
Crystalline V2O5 is widely used in application as chemical sensors, catalysts and solid state batteries.The material possesses an outstanding structural versatility and can be manufactured into nanotubes that have many of the useful properties of the parent crystal significantly enhanced.
0 1 2 3 4 5 6
0
1 0
2 0
3 0
4 0
5 0
6 0
Q (Å -1)
(b )
In
tens
ity (a
.u.)
W a v e ve c to r Q (Å -1)
0
2 0
4 0
6 0
8 0
1 0 0(a )
Q (Å -1)6 9 12 15 18
8
12
6 9 1 2 15 1 8
2
4
V2O5 nanotube
V2O5 crystal
The lack of long range order due to the curvature of the tube walls has a profoundeffect on the diffraction patterns. That of the crystal shows sharp Bragg peaks. The diffraction pattern of the nanotubes has a pronounced diffuse component rendering the traditional techniques for structure determination impossible.
Traditional XRD (reciprocal) vs. Total XRD and PDF (real space)
0 2 4 6 8 10 12 14 16 18 20 22
0
10
20
30
40
50
60
Q(Å-1)
(b)
In
tens
ity (a
.u.)
Wavevector Q(Å-1)
10
20
30
40
50
60 (a)
Q(Å-1)10 11 12 13
1.5
10 11 12 131.0
1.5
2.0
0 2 4 6 8 10 12 14 16 18 20 22
-1
0
1
2
(b)
R
educ
ed s
truct
ure
func
tion
Q[S
(Q)-
1]
Wavevector Q(Å-1)
0
1
(a)
Iel.(Q) Q[S(Q)-1]= [ ] [ ]22. )(/)()( QfcQfcQI iiii
el ∑∑−
0 2 4 6 8 10 12 14 16 18 20-0.6
-0.3
0.0
0.3
0.6
Red
uced
PD
F G
(r)
r (Å)
-0.3
0.0
0.3
0.6
0.9
Bragg peaks:Long-range order Bragg peaks & diffuse scattering * wave vectors Q:
Any-range order & local deviations from it
V2O5 nanotubes - search for a structure model
-0.25
0.00
0.25
-0.25
0.00
0.25
0 5 10 15 20 25-0.25
0.00
0.25
PD
F G
(r)
xerogel V2O5.nH2O model
Radial distance r(Å)
crystalline K2V3O8 model
-0.25
0.00
0.25crystalline V2O5 model
crystalline Zn4V21O58 model
Exp. Data – symbolsModel data – solid line
(a)
(b)
(c)
(d)
V2O5 nanotubes – PDF refinement
The well known 16-atom unit cell of crystalline V2O5 (S.G. Pmmn) fits the experimental data well. The agreement documents the fact the atomic PDF provides a reliable basis for structure determination.
Symbols – exp. dataSolid line – calculated data
Best fit to the experimental PDF data for the nanotubes was achieved on a basis of a 46-atom unit cell (S.G.P⎯1). Even a nanocrystal with the complex morphology of V2O5nanotubes possesses an atomic structure very well defined on the nanometer length scale and well described in terms of a unit cell and symmetry.
V2O5 nanotubes – summary
Structure description of V2O5 nanotubes: Double layers of V-O6 octahedral (green)and V-O4 tetrahedral (red) units are undistorted and stacked in perfect registry with the crystal (a). When bent (b) such layers may form nanoscrolls (c) or closed nanotubes (d).Double layers of such complexity may sustain only a limited deformation. As a result,
V2O5 nanotubes occur with inner diameters not less than 5 nm. The real-size models shown in (c) and (d) have an inner diameter of approx.10 nm and involve 33,000 atoms.The bending of vanadium oxide layers into nanotubes can be explained by the presence of an anisotropy in the distribution of vanadium 4+ and 5+ ions.
More details in Petkov et al Phys. Rev. B 69 (2004) 085410.
CdSe and CdTe nanosize “crystals”
Oleic acid-caped CdSe Thiol-caped CdTeIs there a core-shell “sub-structure” ?
Metallic/semiconductor nanocrystals = Quantum Dots
0 5 10 15 20 25 30-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0
10
20
30
40
50
Inte
nsity
, arb
. u.
Nano CdTe
Red
uced
stru
ctur
e fa
ctor
Q[S
(Q)-1
]
Wave vector Q(Å-1)
Bulk CdTe
Nano CdTe
Bulk CdTe
Nice properties….Not so nice XRD patterns…
In-house vs synchrotron x-ray sources:
0 5 10 15 20 25 30
0.0
0.5
0
10
20
30
40
50
Inte
nsity
, arb
. u.
Nano CdTe
Red
uced
stru
ctur
e fa
ctor
Q[S
(Q)-
1]
Wave vector Q(Å-1)
Bulk CdTe
Nano CdTe
Bulk CdTe
5 10 15 20 25 30
-0.2
-0.1
0.0
0.1
0.2
Ato
mic
PD
F G
(r)
Interatomic distance r(Å)
Bulk CdTe
-0.05
0.00
0.05
0.10 Nano CdTe
Synchrotron (APS): E(x-rays) ~ 90 keV2D detector, ~ a few min, 1D detector – a few hours
In-house: X’Pert, Mo tube E(x-rays) ~ 17 keV1D detector, ~ 48 h
X’Pert
APS
APS
2 3 4 5
0.0
0.1
0.2
0 5 10 15 20 25 30
-0.1
0.0
0.1
0.2
Rwp
= 15 %
Atom
ic P
DF
G(r
)
Interatomic distance r(Å)
Bulk CdTe
0.0
0.1
r(Å)
Nano CdTe
Rwp
= 35 %
CdTe and CdSe nanosize crystals/phase identification
Wurtzite Zinc-blende
Cd-Te Cd-S
Thiol-caped CdTe quantum dots are:i) of zinc-blende type ii) core(CdTe)-shell(CdS) sub-structure
Average crystallite/particle/domain size: Au
The plasmonics of gold nanosize particles has found many applications, ranging from sensors to optical materials. In general, the surface plasmon resonance is controlled by many factors, including particle’s size and structure. That is why detailed knowledge about the 3D structure is a prerequisite to understanding and possibly improving the useful properties of Au nanosize particles. We studied dendrimer stabilized Au particles of size approx. 3 nm, 15 nm and 30 nm. TEM images are shown above.
Au
0 10 20 30 40 500
2
4
6
8 Bulk Au
3 nm
Dendrimer stabilized Au nanoparticles in water
15 nm
Atom
ic P
DF
G(r
)
Radial distance r(Å)
30 nm
Experimental atomic PDFs (symbols) for Au nanosize particles. The x-ray diffraction experiments were carried out at the beamline11IDC at the Advanced Photon Source using x-rays of energy 115 keV.
The PDF for bulk gold shows well defined peaks to very long real space distances reflecting the presence of a 3D periodicity and long-range order in this crystalline material. Theexperimental data are well fit by a model based on the face centered cubic (fcc) structure of crystalline gold (solid line in red). The experimental PDFs for the gold nanosize particles decay to zero at much shorter real space distances reflecting the reduced length of structural coherence/size. As can be expected this length diminishes with the particle’s size.Therefore: we can use the real space distance at which PDF decays to zero as an estimate for the length of structural coherence/nanoparticle/nanodomain size.
Average crystallite/particle/domain size: Au
“Particle/Crystallite/Domain”sizedistribution
Conclusion:
Total XRD & PDF analysis can yield the 3D structure of “unusual” crystals in detail. The approach succeeds because it relies on total scattering data, including both Bragg-like and diffuse scattering. It probes the bulk (not the surface like TEM/imaging) over the entire length (not only the first coordination sphere like EXAFS/spectroscopy) of structural coherence/order materials show. It is flexible with respect to sample’s state, morphology, amount, phase homogeneity and environment. It may be used for “phase” identification as well as to obtain estimates for “crystallite/domain” size and “lattice strain”.
X-ray crystallography without “usual” crystals is possible !How: by employing “unusual/non-traditional” approaches !
More info: WWW resources
http://www.phy.cmich.edu/people/petkov/nano.html
Acknowledgments:
• Funding: NSF, ARL, DOE….• Facilities: CHESS, NSLS, APS….• XRD manufacturers : PANalytical• Post-docs/Grad students: M. Gateshki, Y. Peng, S.
Pradhan….. • Beamline scientists: Stefan Kycia (A2), Tom Vogt (x7a),
Sarvjit Shastri & Peter Lee (ID-1), Doug Robinson (6-ID),Yang Ren (11-ID-C), Peter Chupas (11-ID-B)……
• Sample makers/collaborators: too many to list… Thank you all !