^ z } } µ } v y r Ç ] ( ( ] } v v o Ç ] } ( z ] v ( ] o u v µ ( w courses... · 2019-04-11 ·...

Short Course onX-ray diffraction analysis of thin films and surfaces: i. Glancing/grazing incident XRD (GIXRD)ii. Residual stress analysis (RSA)

Pio John Buenconsejo (Dr)FACTS Scientisthttp://research.ntu.edu.sg/facts

5/March/2019 FACTS short course, Buenconsejo 1

Content

Background: X-ray analysis of thin film materials Glancing/grazing incident X-ray

diffraction Residual stress analysis by XRD Summary


What is a thin film sample?

Prepared by: Coating or deposition Chemical or Physical reaction

Characteristics: Thickness, roughness, density,

porosity, residual stress, topology, structures

Crystallinity, orientation, ordering, composition, defects, epitaxial strain

Applications: Electronic devices Sensors & MEMS devices Surface science & engineering

Film 1 mm to 10 mm Thin film 10 nm to 1 mmUltra thin film 10 nmSurface or monolayer < 1 nm

Thin film

Substrate


X-ray interaction with thin films

Incident X-rays are reflected/scattered/diffracted:• X-ray reflectivity (XRR) • X-ray scattering (SAXS, GISAXS)• X-ray Diffraction (PXRD, GIXRD

GIWAXS, HRXRD)

Incident X-rays excites the sample to produce signals:• X-ray Fluorescence (XRF)• X-ray Photon Emission (XPS)

Thin film

Substrate

Incident X-rays• Laboratory source• Synchrotron source


X-ray analysis of thin film materials

Amorphous Non-crystalline Thickness Roughness Density Nano-structure

Thin film

Substrate

Polycrystalline Crystalline, Phase ID Randomly oriented Thickness Roughness Density Nano-structure Residual stress

Textured Crystalline, Phase ID Preferred

orientation Thickness Roughness Density Nano-structure Residual stress

Epitaxial Single crystal Orientation Thickness Roughness Density Nanostructure Epitaxy, strain,

defects, composition

PXRD or TFXRDPFHRXRD XRRGISAXS/WAXS

RSA


X-ray analysis of thin films and surfaces in FACTS

Instrument PXRDTFXRD PF XRR GISAXS

GIWAXS RSA HRXRD

XRD Cluster 1Shimadzu XRD

XRD Cluster 2Bruker D8 AdvancePanalytical X’pert Pro

Bruker D8 Discover* Xenocs Nanoinxider**

*In-situ heating RT to 1100C **In-situ cooling-heating, -150C to 350C Limited capability

PXRD Powder XRD (Bragg-Brentanno, 2q-q scan)TFXRD Thin film XRD (Glancing/grazing angle, 2q scan)PF Pole figure (phi-,chi-scan)XRR X-ray reflectometry (specular and off-specular)GISAXS Grazing Incidence Small Angle X-ray Scattering (in/out-of-plane)GIWAXS Grazing Incidence Wide Angle X-ray Scattering (in/out-of-plane)RSA Residual stress analysis (sin2y, multiple-hkl, whole pattern decomposition)HRXRD High resolution XRD (2q-w, w-2q, rocking curve, phi-, chi-scan)


PXRD and TFXRDWhat’s the difference?


PXRD (q/2q) versus TFXRD (GIXRD)

T. Ryan, Journal of Chemical Education 79(5) 2001, 613-616

1. Why is there a difference in diffraction geometry?2. How much absorption and penetration depth?

Bragg-Brentanno: q/2q scanGIXRD: 2q scan at fixed q or a


q/2q diffraction geometry

Scattering plane

QDiffraction/scattering vector

XD

q 2q

Scattering plane

(hkl)1 (hkl)2 (hkl)3

nl =2dsinq Bragg condition Scattering/Diffraction vector is

always normal to the surface q/2q is a symmetric scan hkl parallel to the surface Penetration depth is not controlled Coplanar configuration


q/2q : Bragg-Brentanno geometry

Powder XRD: Bragg-Brentanno (parafocusing) geometry Divergent beam optics:

- Including crystals with tilted hkl normal Angular resolution can be improved with narrow

slits

hkl

Diffraction vectorX D

2qq Div. slit

Bragg-Brentanno geometry with optics

Focus length


q/2q: Bragg-Brentanno geometry

X D

Goniometer circle

Focusing circle

q 2q

in-focusout-of-focus out-of-focus

R= 320 mm (dashed curves)= 240 mm (solid curves)


q/2q GIXRD diffraction geometry

X D

Goniometer circle

Focusing circle

q 2q X

D

Focusing circle

Goniometer circle

a

2q-a

Only one point is in focus Requires narrow/parallel incident beam 5/March/2019 FACTS short course, Buenconsejo 12

GIXRD diffraction geometry

Thin film XRD: Glancing or Grazing incident angle Asymmetric scan, non-parafocusing Incident/diffracted parallel beam Controlled penetration depth Sample surface flat alignment is important

(surface & height) Diffraction vector is not parallel to the sample

surface normal

hkl

Diffraction vector

X

D

2q

a

a

Scattering plane


Beam footprint with parallel beam on the sample


Beam footprint in GIXRD (parallel beam)

Beam footprint (B):

a

BW = beam width

0 1 2 3 41

10

100

1000

B (

mm

)

a (o)

Beam width 0.1 mm 0.2 mm 0.6 mm 1 mm


Goebel mirror


Examples: GIXRD versus BBG scan

40 60 80 1001000

10000

100000

1000000

1E7

1E8

1E9

1E10

1E11

1E12

1E13

0

20000

40000

60000

80000

100000

Log

Inte

nsi

ty

2q (o)

q-2q scan substrate

(222)fcc

(311)fcc(220)

fcc

(200)fcc

Inte

nsity

GIXRD a =1o

63Au-35Cu-2Al thin film fcc-phase

t = 681 nm

Substrate: SiO2/Si-001

(111)fcc GIXRD:

Probes the thin film region only Relative Intensity is lower

Bragg-Brentanno (q/2q scan): Additional contribution from the

substrate Very high intensity

How much difference in the absorption and penetration depth?


X-ray absorption and penetration depth

Lambert-Beer law (X-ray absorption):

µ = linear absorption coefficient = µmrµm = mass absorption coefficient r = mass densityl = X-ray path length

Penetration depth⁄ (normal incidence)⁄ (non-normal incidence)

X-ray absorption effects on materials: X-ray absorption follows Lambert-Beer law which relates the attenuation of light

(electromagnetic radiation) travelling through a material to the material’s properties. Penetration depth is the depth at which the intensity of radiation incident on a

material is attenuated to 1/e (37%).


q/2q scan: Absorption factor

Ref: Thin Film analysis by X-ray Scattering, Birkholtz

Io Io exp(-2ml)

zl l

q q

t𝑤ℎ𝑒𝑟𝑒: 𝑙 =

𝑧

𝑠𝑖𝑛𝜃

/ Absorption factor:

A = ∫ ( )

∫ ( )

Intensity is attenuated by:

Solution of the integral:

Infinitely thick sample:


q/2q: Absorption factor

Absorption factor:

/


Diffraction intensity from a polycrystalline sample:


q/2q: Penetration depth


𝟏𝒆⁄

𝟏𝝁 normal incidence

Penetration depth


GIXRD: Absorption factor

Thin Film analysis by X-ray Scattering, Birkholtz

Io

Io exp(-ml)

zl1

l2

a2q - a

t

X-ray path length:

𝑘 =1

𝑠𝑖𝑛𝛼+

1

sin (2𝜃 − 𝛼)


GIXRD: X-ray penetration depth

Thin Film analysis by X-ray Scattering, Birkholtz

/

1. Attenuation of X-rays to 1/e of its initial value

2. t63 for AGIXRD = 1-1/e (mt63ka = 1, 63%) May exceed the thickness of a thin layer

3. Information depth, based on weighted average over the sampling depthfor infinitely thick-layer it approaches t63


GIXRD: X-ray penetration depth

40 60 80 1001000

10000

100000

1000000

1E7

1E8

1E9

1E10

1E11

1E12

1E13

0

20000

40000

60000

80000

100000

Log

Inte

nsi

ty

2q (o)

q-2q scan substrate

(222)fcc

(311)fcc(220)

fcc

(200)fcc

Inte

nsity

GIXRD a =1o

63Au-35Cu-2Al thin film fcc-phase

t = 681 nm

Substrate: SiO2/Si-001

(111)fcc

a=1o

depth =55 nm

q > 15o

depth > 900 nm

---Note-----i) Porosity, columnar grains, voids, defects in the material could reduce the effective density of the film and therefore the penetration depth could be deeper. ii) Thin film density could be measured using XRR.


Useful links for X-ray penetration depth

Online penetration depth calculator:http://gixa.ati.tuwien.ac.at/tools/penetrationdepth.jsf

Absorption coefficient reference:https://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.htmlhttps://physics.nist.gov/PhysRefData/FFast/html/form.html


GIXRD at a ~ ac

Near ac: Very surface sensitive GIXRD (in-plane

scattering/diffraction)

/

/

4. Penetration depth:

Absorption limited

Total reflection


GIXRD: Out-of-plane scattering/diffraction

Scattering plane


X

D

a2q-a

a

Scattering plane


a is constant a ≈ arcsin(mt) Thickness/depth information

profile (microstructure, strain, phase)

Coplanar configuration GISAXS, GIWAXS out plane

scattering


GIXRD: In-plane scattering/diffraction

QX

ai

afScattering plane D

ai

(hkl)1

af

(hkl)2

Scattering plane

Scattering plane is almost parallel to the surface

ai, af are constant and very small almost near the surface plane

Evanescent wave (few nm of the surface) depth information: ai a~arcsin(mt) hkl-planes perpendicular to the surface GISAXS, GIWAXS, -inplane scattering


GIXRD: 2D diffraction


Xa

qz

(+)qx

qz

(+)qx

(-)qx

2D detector: Captures the out-of-plane

and in-plane diffraction Good for GISAXS/GIWAXS

experiments Anisotropy or preferred

orientation

x

y

zD

D


GIXRD: 2D diffraction


Xa

qz

(+)qx

qz

(+)qx

(-)qx

2D diffraction: Makes sense with

point X-ray source

x

y

z

This will not work for a Line beam X-ray source5/March/2019 FACTS short course, Buenconsejo 30

2D diffraction: GIWAXS in-plane and out-of-plane example

More of this technique in GISAXS/GIWAXS short course


Residual stress analysis by X-ray Diffraction


Residual stress in materialsOrigin: Thin-film/substrate

difference in mechanical properties multi-layer stack

Surface deformation nanocrystallization or amorphization by mechanical deformation

Micro(nano)-structures porosity, defects 2D or 3D structures Composite structures

Extrinsic or Intrinsic

Why do we need to quantify it? Reliability & performance Processing & product development Functional properties Failure analysis


Residual stress

Macrostress: Average over many grains sI Diffraction peak shifts sin2y method and its

variants

Microstress: Grain-specific, internal sII & sIII Diffraction peak shift and

broadening Diffraction peak shape

analysis: • Williamson-Hall (Acta

Metall. 1 (1953) 22).• Warren-Averbach• Whole pattern fitting


X-ray residual stress analysis

Specimen

homogeneousInhomogeneous

polycrystallineSingle-crystalline

Isotropic material

Quasi-isotropic

Anisotropic

Simple Mechanical constants (E, v)

Relatively simple hkl dependent elastic constants X-ray elastic constants (XEC) Complicated Texture, grain-interaction models X-ray stress factors (XSF)

Data collection?Data analysis?


Basic elasticity equations

: strain: stress

: compliance tensor: elastic stiffness

s11

s12

s13

s31

s32

s33

s21

s22

s23

Stress tensors in the sample reference frame

Hooke’s law

Isotropic materials:

E (Young’ modulus) and n (Poisson’s ratio) are materials dependent mechanical constants


Elastic constants__________________________________________Triclinic None (or center of symmetry)Monoclinic 1 twofold rotationOrthorhombic 2 perpendicular twofold rotationTetragonal 1 fourfold rotation around [001]Rhombohedral 1 threefold rotation around [111]Hexagonal 1sixfold rotation around [0001]Cubic 4 threefold rotations around <111>

Anisotropy factor

Cubic



SS3

SS1

SS2

L2

L1

L3y

f

Sample reference frame: SS1 SS2 SS3 SS3 = Surface normal SS1 & SS2 are in-plane (surface)

directions

Laboratory reference frame: L1 L2 L3 𝐿 = 𝐿 = Diffraction vector

𝜀 = 𝜀

f in-plane rotation angle

y azimuth angle

Tensor transformation applied to express the strain (stress) in the sample coordinate system

Fundamental equation for x-ray residual stress analysis:

𝜀 =1

2𝑆 𝜎 𝑐𝑜𝑠 𝜙 + 𝜎 𝑠𝑖𝑛 𝜙 + 𝜎 𝑠𝑖𝑛2𝜙 𝑠𝑖𝑛 𝜓 + 𝜎 𝑐𝑜𝑠𝜙 + 𝜎 𝑠𝑖𝑛𝜙 𝑠𝑖𝑛2𝜓 + 𝜎 𝑐𝑜𝑠 𝜓

+𝑆 (𝜎 + 𝜎 + 𝜎 )



SS1

SS2

SS3𝝓𝝍y

f


𝜀 = 𝜀 =1


+𝑆 (𝜎 + 𝜎 + 𝜎 )

𝐵𝑟𝑎𝑔𝑔 𝑠 𝑙𝑎𝑤: 𝜆 = 2𝑑 𝑠𝑖𝑛𝜃

= Diffraction vector

𝒅𝒉𝒌𝒍

s11

s12

s13

s31

s32

s33

s21

s22

s23

X-ray elastic constants:


How do we collect RSA data?

Conventional diffraction geometry: y: angle between sample (S3) and

diffractometer (L3) Can be done as c or w tilt or both Single hkl (set at the Bragg angle 2qhkl)2q

L3//SS3

x d

y = 0

q

w/q

L3SS3 y

2q

x d

q

w/q

c tilt

c=y

2q

x d

q

w/q

L3 SS3

yw tilt

y

w





L3//SS3

x d

y = 0

q

w/q

2q

x d

q

w/q

L3 SS3

yw tilt

y

ww

Scattering plane

(-) y

SS3y

L3

w w

SS3 y

L3

SS3 L3

(0) y (+) y Maximum y tilt is limited by q Penetration depth is not controlled 2q-w scan mode5/March/2019 FACTS short course, Buenconsejo 41




L3//SS3

x d

y = 0

q

w/q

y-tilt is not limited by q Chi (c) scan in a Eulerian cradle y-tilt in GIXRD is possible (fixed a),

penetration depth is controlled

L3SS3 y

2q

x d

q

w/q

c tilt

c=y

L3

SS3

L3

SS3

L3

SS3

(+)y (0)y (-)y


(+)qx

(-)qx

qz




L3//SS3

x d

y = 0

q

w/q

2D-diffraction simultaneously captures various y-tilt (c azimuth)

GIXRD, penetration depth controlled

L3SS3 y

2q

x d

q

w/q

c tilt

c=y

SS3

(+)y (-)y

L3

(+)y



Grazing incident geometry: y: angle between sample (S3) and

diffractometer (L3) y tilt by 2q scan multiple hkl (as set by the Bragg angle

2qhkl) 2D diffraction will require no tilting Penetration depth is controlled

2q

SS3

x

d

a

w/q

yL3

Grazing incident @ fixed a

2q

SS3

x

d

a

w/q

yL3

2q scan

a

Scattering plane


SS3y

L3

SS3yL3

SS3y

L3

a a

y-tilt by 2q scan will require multiple hkl5/March/2019 FACTS short course, Buenconsejo 44

FACTS XRD

2qq/w

SS3

𝝓𝝍y = 0

w

Scattering plane

(-) y

SS3y

L3

w w

SS3 y

L3

SS3 L3

(0) y (+) y

q/w2q

SS3𝝓𝝍

D8 Discover: Horizontal goniometerD8 Advance: vertical goniometer

Eulerian cradle

w-tilt c-tilt GIXRD f-rotation

w-tilt


Residual stress data analysis

I. Single hkl• Triaxial, biaxial, uniaxial

II. Multiple hkl• g(y, hkl), f(y, hkl)

III. General least-squares analysis of any stress state• Non-linear least square fitting algorithms

e

sin2y

e

sin2y

y>0

y<0 e

sin2y

X-ray Elastic Constant:

sin2y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

y 18.43 26.57 33.21 39.23 45 50.77 56.79 63.43 71.57

X-ray Stress Factor:


𝜀 = 𝜀 =1


+𝑆 (𝜎 + 𝜎 + 𝜎 )

variation of y at constant f


Non-linear: Textured materials Gradient stress Shear stress

SS1

SS2

efyy

f

SS3

Single hkl analysis

112 114 116 118 1200

200

400

600

800

1000

1200

1400

1600

1800

2000

+y

2q

-y

f = 0

112 114 116 118 120

2q

f = 45

112 114 116 118 120

2q

f = 90

Material: Au71Cu25Al4 thin filmfcc-AuCu(Al) phasethickness = 725 nm

hkl (331) at 2q = 116.23

Young’s modulus = 79 GPaPoisson’s ratio = 0.44Arx = 2.8

S1 = -4.153E-6; 1/2S2 = 1.398E-5


𝜀 = 𝜎 𝑐𝑜𝑠 𝜙 + 𝜎 𝑠𝑖𝑛 𝜙 + 𝜎 𝑠𝑖𝑛2𝜙 𝑠𝑖𝑛 𝜓 + 𝜎 𝑐𝑜𝑠𝜙 + 𝜎 𝑠𝑖𝑛𝜙 𝑠𝑖𝑛2𝜓 + 𝜎 𝑐𝑜𝑠 𝜓

+𝑆 (𝜎 + 𝜎 + 𝜎 )


Single hkl analysis

0.0 0.2 0.4 0.6 0.8 1.0-0.004

-0.002

0.000

0.002

0.004-0.004

-0.002

0.000

0.002

0.004

0.0 0.2 0.4 0.6 0.8 1.0

-0.004

-0.002

0.000

0.002

0.004

stra

in

sin2y

f = -90

f = -45

stra

in

f = 0 -y +y

stra

in

Stress tensor (MPa) =

Triaxial stress analysis:

s11

s12

s13

s31

s32

s33

s21

s22

s23

Principal stress tensor (MPa) =

Biaxial stress!5/March/2019 FACTS short course, Buenconsejo 48

Single hkl analysis: sin2y variants

𝜎 =𝜎 0 00 𝜎 00 0 0

𝜀 = 𝑆 𝜎 𝑠𝑖𝑛 𝜓 + 2𝑆 𝜎

𝜎 =𝜎 𝜎 0𝜎 𝜎 00 0 0

𝜀 = 𝑆 𝜎 𝑠𝑖𝑛 𝜓 + 𝑆 (𝜎 +𝜎 )


𝜀 = 𝑆 + 𝜎 𝑠𝑖𝑛 𝜓 + 𝑆 (𝜎 +𝜎 )

𝜎 =𝜎 0 00 0 00 0 0

𝜀 = 𝑆 𝜎 𝑠𝑖𝑛 𝜓 + 𝑆 𝜎Uniaxial

𝜎 =𝜎 0 00 𝜎 00 0 0



Biaxial

Biaxial Rotationally symmetric

Biaxial principal stress

Triaxialprincipal stress 𝜎 =

𝜎 0 00 𝜎 00 0 𝜎

𝜀 = 𝑆 𝜎 − 𝜎 𝑠𝑖𝑛 𝜓 + 𝑆 (𝜎 +𝜎 + 𝜎 )+ 𝑆 𝜎

𝜀 = 𝑆 (𝜎 +𝜎 + 𝜎 )+ 𝑆 𝜎

𝜀 = 𝑆 𝜎 − 𝜎 𝑠𝑖𝑛 𝜓 + 𝑆 (𝜎 +𝜎 + 𝜎 )+ 𝑆 𝜎


Multiple hkl analysis

Material: Au71Cu25Al4 thin filmfcc-AuCu(Al) phasethickness = 725 nm

Young’s modulus = 79 GPaPoisson’s ratio = 0.44Arx = 2.8

40 60 80 100 12010

100

1000

10000

100000

a = 20

420

331

400

22231

1220

111

Inte

nsity

2q

200

a = 0.5

-0.000006 -0.000004 -0.000002 0.000000 0.000002

-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

stra

in

f (y, hkl)

a =0.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40400

500

600

700

800

900

1000

1100

Axial stress

Axi

al s

tres

s

thickness, mm

GIXRD


Multiple hkl analysis

g(y, hkl) method:Biaxial stress state 𝑔 𝜓, ℎ𝑘𝑙 =

𝑆𝑠𝑖𝑛 𝜓

= 𝑔 𝜓, ℎ𝑘𝑙 𝜎 + (𝜎 +𝜎 )

= 𝑔 𝜓, ℎ𝑘𝑙 𝜎 + (𝜎 +𝜎 )

= 𝑔 𝜓, ℎ𝑘𝑙 + 𝜎 + 𝜎 +𝜎

f(y, hkl) method:Rotationally symmetric Biaxial stress state

𝑓 𝜓, ℎ𝑘𝑙 = 2𝑆 + 𝑠𝑖𝑛 𝜓 𝜀 = 𝑓 𝜓, ℎ𝑘𝑙 𝜎


General least-square analysis of any stress state

wi is weighting factor ss is a fitting parameter

Most general solution Requires more computing power Non-linear least squares fitting Can combine single hkl and multiple hkl

data Works well with textured (highly

anisotropic) samples For highly anisotropic samples it requires

use of grain-interaction models (XSF)


Summary

q/2q scan vs GIXRD geometry X-ray absorption & penetration depth in-plane & out of plane diffraction

Residual stress analysis background on residual stress how to collect the data how to analyse the data

Suggested reading materials: “Residual Stress (Measurement by Diffraction and Interpretation)” Noyan, Cohen, 1987 Springer

“Thin film analysis by X-ray Scattering” Mark Birkholz

“Stress analysis of polycrystalline thin films and surface regions by XRD” Wezel, et. al. J. Appl. Crys. (2005) 38 1-29


^ z } } µ } v y r Ç ] ( ( ] } v v o Ç ] } ( z ] v ( ] o u v µ ( w courses... · 2019-04-11 ·...

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