X-ray crystallography without crystals · 2007-08-27 · X-ray crystallography without crystals...

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


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.


Lattice parameters

a b c - sidesα β γ - angles

cb

a


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


L-shell

III

III

Kα2 Kα1

K-shell


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.


• 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


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


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.


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


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


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


111

05-05-2005


111

220

311

05-05-2005


111

220

311

004 331

05-05-2005


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

Phase ID – Depth profiling

Cu(In,Ga)Se2 solar cells

GIXRD - Thin film phase analysis


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 [email protected]

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

PDF

G(r)

Radial distance [A-]

PDF

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


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


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

PDF

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 !

X-ray crystallography without crystals · 2007-08-27 · X-ray crystallography without crystals...

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Transcript of X-ray crystallography without crystals · 2007-08-27 · X-ray crystallography without crystals...