Combining X-ray and TEM techniques for microstructure studies on …€¦ · Combining X-ray and...
Transcript of Combining X-ray and TEM techniques for microstructure studies on …€¦ · Combining X-ray and...
Combining X-ray and TEM techniques for microstructure studies on thin films and thin film nanocomposites
David Rafaja, Institute of Materials [email protected]
Motivation
2
Thin films of molybdenum deposited on glass substratesUsed as contacts in the Cu(In,Ga)Se2 solar cells, etc.
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0 50 100 150 200 250 300 3505
10
15
20
25
Resis
tivity (c
m)
Temperature (°C)
RF magnetron
DC magnetron
Pulsed DC magnetron
Intrinsic resistivity of Mo
D. Rafaja, H. Köstenbauer, U. Mühle, C. Löffler, G. Schreiber, M. Kathrein, J. Winkler, Thin Solid Films 528 (2013) 42-48.
Motivation
3
Drude model of electrical conductivity
eE
mvm
eEv
tm
vv
eEvdt
dvm
max
max
max exp1
t
v
vmax
Microstructure defects act as scattering centers for electrons
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-
3D
2D
1D
0D
Microstructure defects
4
Structural vacancies
Foreign atoms
Dislocations
Stacking faults
Twin boundaries
Grain boundaries
Interfaces & surfaces
Local phase transitions
Precipitates of other phases
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XRD versus TEM
5
X-ray diffraction
Integral measurement of atomic positions, interplanar distances and their variations
Transmission electron microscopy
Local measurement of atomic positions, interplanar distances and their variations
𝐼 = 𝐼0
𝑛
𝑓𝑛 exp 𝑖 𝑞 ∙ 𝑟′𝑛
2
= 𝐼0
𝑛
𝑓𝑛 exp 𝑖 𝑞 ∙ 𝑟𝑛 + 𝑢𝑛
2
= 𝐼0
𝑛
𝑓𝑛 exp 𝑖 𝑞 ∙ 𝑟𝑛 exp 𝑖 𝑞 ∙ 𝑢𝑛
2
Diffracted intensity
Line positions
𝑞 =2𝜋
𝑑=
4𝜋
𝜆sin 𝜃
Line shift
Δ 𝑞 = −2𝜋Δ𝑑
𝑑2 = −4𝜋
𝜆sin 𝜃
Δ𝑑
𝑑= − 𝑞 𝜀
Line broadening
Δ 𝑞 212 =
4𝜋
𝜆sin 𝜃
Δ𝑑
𝑑
212
= 𝑞 𝜀212
Structure factor, line positions
Diffuse scattering, line broadening, …
Point defects as seen by XRD
6
Produce diffuse scattering
Affect stress-free lattice parameters
But
The measured lattice parameters are affected by:
Foreign atoms & atomic vacancies (Vegard-like dependence)
Residual stresses (Lattice deformation depending on the macroscopic direction)
Stacking faults (Change of the interplanarspacing depending on the diffraction indices)
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How to determine the stress-free lattice parameters?
7
7
I. C. Noyan and J. B. Cohen: Residual Stress Measurement by Diffraction and Interpretation, Springer Verlag, New York, Berlin, Heidelberg, 1987.
A.J. Perry, V. Valvoda, D. Rafaja, Thin Solid Films 214 (1992) 169-174.
sin²
Lattic
e d
efo
rmation
/E
2/(+1)
The sin² plot
For known X-ray elastic constants and stress-free interplanar spacing (d0)
For known Poisson ratio, Young modulus and d0
Materials under uniaxial stress
0
First, the effect of the residual stress must be described
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𝜀𝜓ℎ𝑘𝑙 =
𝑑𝜓ℎ𝑘𝑙 − 𝑑0
ℎ𝑘𝑙
𝑑0ℎ𝑘𝑙
= 𝜎1
2𝑠2ℎ𝑘𝑙 sin² 𝜓 + 2𝑠1
ℎ𝑘𝑙
𝜀𝜓ℎ𝑘𝑙 =
𝜎
𝐸ℎ𝑘𝑙1 + 𝜈ℎ𝑘𝑙 sin² 𝜓 − 2𝜈ℎ𝑘𝑙
𝑠1ℎ𝑘𝑙 ≡ −
𝜈ℎ𝑘𝑙
𝐸ℎ𝑘𝑙
12𝑠2
ℎ𝑘𝑙 ≡1 + 𝜈ℎ𝑘𝑙
𝐸ℎ𝑘𝑙
How to determine the stress-free lattice parameters?
8
I. C. Noyan and J. B. Cohen: Residual Stress Measurement by Diffraction and Interpretation, Springer Verlag, New York, Berlin, Heidelberg, 1987.
A.J. Perry, V. Valvoda, D. Rafaja, Thin Solid Films 214 (1992) 169-174.sin²
Inte
rpla
na
rsp
acin
g
Stress-free
interplanar
spacing
/E
2/(+1)
The sin² plot For known X-ray elastic constants
For known Poisson ratio and Young modulus
Materials under uniaxial stressMeasurements on one family of lattice planes
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𝑑𝜓ℎ𝑘𝑙 = 𝑑0
ℎ𝑘𝑙 𝜎1
2𝑠2ℎ𝑘𝑙 sin² 𝜓 + 2𝑠1
ℎ𝑘𝑙 + 1
𝑠1ℎ𝑘𝑙 ≡ −
𝜈ℎ𝑘𝑙
𝐸ℎ𝑘𝑙
12𝑠2
ℎ𝑘𝑙 ≡1 + 𝜈ℎ𝑘𝑙
𝐸ℎ𝑘𝑙
𝑑𝜓ℎ𝑘𝑙 = 𝑑0
ℎ𝑘𝑙𝜎
𝐸ℎ𝑘𝑙1 + 𝜈ℎ𝑘𝑙 sin² 𝜓 − 2𝜈ℎ𝑘𝑙 + 1
9
How to determine the cubic stress-free lattice parameter?
I. C. Noyan and J. B. Cohen: Residual Stress Measurement by Diffraction and Interpretation, Springer Verlag, New York, Berlin, Heidelberg, 1987.
A.J. Perry, V. Valvoda, D. Rafaja, Thin Solid Films 214 (1992) 169-174.sin²
Lattic
e p
ara
mete
rs
Stress-free
lattice
parameter
/E
2/(+1)
The sin² plot For known X-ray elastic constants
For known Poisson ratio and Young modulus
Cubic materials under uniaxial stress
Measurements on different families of lattice planes (GAXRD)
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𝑎𝜓ℎ𝑘𝑙 = 𝑎0 𝜎
1
2𝑠2ℎ𝑘𝑙 sin² 𝜓 + 2𝑠1
ℎ𝑘𝑙 + 1
𝑠1ℎ𝑘𝑙 ≡ −
𝜈ℎ𝑘𝑙
𝐸ℎ𝑘𝑙
12𝑠2
ℎ𝑘𝑙 ≡1 + 𝜈ℎ𝑘𝑙
𝐸ℎ𝑘𝑙
𝑎𝜓ℎ𝑘𝑙 = 𝑎0
𝜎
𝐸ℎ𝑘𝑙1 + 𝜈ℎ𝑘𝑙 sin² 𝜓 − 2𝜈ℎ𝑘𝑙 + 1
10
Anisotropy of elastic constants
40 60 80 100 120
0
5000
10000
Inte
nsity (
co
un
ts)
Diffraction angle (°2)
36 38 40 42 44 62 64 66
0
5000
10000
Inte
nsity (
co
un
ts)
Diffraction angle (°2)
11
1
20
0
22
0
31
1
22
2 40
0
33
1
42
0
11
1
20
0
22
0
Rietveld refinement using MAUD (L. Lutterotti): http://www.ing.unitn.it/~maud/M. Ferrari, L. Lutterotti, J. Appl. Phys. 76 (11) (1994) 7246-7255.
- Residual stress
- Anisotropy of elastic constants
- Preferred orientation of crystallites (inclined texture)
CuK radiation
CAE (Cr,Al,Si)N
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11
Calculation of the cubic stress-free lattice parameter
I.C. Noyan, J.B. Cohen: Residual stresses - Measurement by diffraction and interpretation, Springer, New York, 1987.
V. Valvoda, R. Kužel, R. Černý, D. Rafaja, J. Musil, S. Kadlec, A.J. Perry, Thin Solid Films 193/194 (1990) 401.
C. Kral, W. Lengauer, D. Rafaja, P. Ettmayer, J. Alloys Comp. 265 (1998) 215.
0.0 0.2 0.4 0.6 0.8 1.04.11
4.12
4.13
4.14
4.15
4.16
4.17
11
12
00
22
0
31
12
22
40
0
33
14
20
42
2
33
3
La
ttic
e p
ara
me
ter
(Å)
sin2
a0
slope =
= 0.32 sin²0 = 0.48 a0 = 4.1391 Å
Isotropic cubic materials (sin² method):
1
2sin 0
2
CAE (Cr,Al,Si)N
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𝑎𝜓 = 𝑎0
𝜎
𝐸1 + 𝜈 sin²𝜓 − 2𝜈 + 1
1 + 𝜈
𝐸𝜎𝑎0
Anisotropy of the elastic constants
12
A. Reuss, Z. angew. Math. Mech. 9 (1929) 49.E. Kröner, Z. Physik, 151 (1958) 504.R.W. Vook and F. Witt, J. Appl. Phys. 36 (1965) 2169.
Elastic constants of -Fe
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13
Anisotropy of the elastic lattice deformation
0.0 0.2 0.4 0.6 0.8 1.00.411
0.412
0.413
0.414
0.415
0.416
0.417
111
200
220
311
222
400
331
420
422
333
Lattic
e p
ara
mete
r (n
m)
sin2
CAE (Cr,Al,Si)N
12sin 1
2
221
0 hkhkhk ssaa
2222
222222
2221
221
111
1
22
2
kh
hkkh
BAE
s
BAE
s
hk
hkhk
hk
hkhk
For uni-axial residual stress
D. Rafaja, M. Dopita, M. Růžička, V. Klemm, D. Heger, G. Schreiber, M. Šíma, Surf. Coat. Technol. 201 (2006) 2835. D. Rafaja, C. Wüstefeld, J. Kutzner, A.P. Ehiasarian, M. Šíma, V. Klemm, D. Heger, J. Kortus, Z. Krist. 225 (2010) 599-609.
hks1
hks2
001
01
2
0221
2
0221
2
2
sin
sin
aaA
aB
aB
aAahk
… slope
… scatter
… shift
Lattice parameter
14
Anisotropy of the elastic lattice deformation
0.0 0.2 0.4 0.6 0.8 1.04.11
4.12
4.13
4.14
4.15
4.16
4.171
11
20
0
22
0
31
12
22
40
0
33
14
20
42
2
33
35
11
La
ttic
e p
ara
me
ter
(Å)
sin2
100 111
R.W. Vook, F. Witt, J. Appl. Phys. 36 (1965) 2169.
D. Rafaja, V. Valvoda, R. Kužel, A.J. Perry, J.R. Treglio, Surf. Coat. Technol. 86-87 (1996) 302-308.
D. Rafaja, M. Dopita, M. Růžička, V. Klemm, D. Heger, G. Schreiber, M. Šíma, Surf. Coat. Technol. 201 (2006) 2835-2843.
F. Attar, T. Johannesson, Thin Solid Films 258 (1995) 205.
00101
2
02212
0221
2221
221
111
1
2
221
0
22sinsin
1
22
2
12sin
aaAaBaBaAa
BAE
s
BAE
s
ssaa
hk
hk
hkhk
hk
hkhk
hkhkhk
Anisotropy of the (cubic) elastic constants:
2222
222222
kh
hkkh
100
111
slope scatter shift
200 = 0.214 sin²0 = 0.353 a0 = 4.1391 Å
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15
Anisotropy of the elastic constants
0.0 0.2 0.4 0.6 0.8 1.04.11
4.12
4.13
4.14
4.15
4.16
4.171
11
20
0
22
0
31
12
22
40
0
33
14
20
42
2
33
35
11
La
ttic
e p
ara
me
ter
(Å)
sin2
J.F. Nye: Physical Properties of Crystals – Their Representation by Tensors and Matrices, Clarendon Press, Oxford, 1985.
I.C. Noyan, J.B. Cohen: Residual stresses - Measurement by diffraction and interpretation, Springer, New York, 1987.
D. Rafaja, M. Dopita, M. Růžička, V. Klemm, D. Heger, G. Schreiber, M. Šíma, Surf. Coat. Technol. 201 (2006) 2835-2843.
122sin3sin
3;
12sin
12000
2
00
2
12110
4421
12110
01211221
0121
1
2
221
0
SaSaSaSSaa
SSSS
SSSsSSs
ssaa
hk
hkhk
hkhkhk
Anisotropy of the cubic elastic constants (Reuss approach):
2222
222222
kh
hkkh
100
111
1211
44
12110
00
2
21
SS
S
SSa
SaA
A = 3.0
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16
Elastic compliance
4421
1211001211221
0121
001
2
221
0
;3;
12sin
SSSSSSSsSSs
afassaa
hkhk
hkhkhk
S matrix from the ab initio calculation:
- abinit code (DFT using PAW) for calculation of the atomic positions
- stress-strain approach for calculation of the elastic constants
440000000
044000000
004400000
00020954-54-
00054-20954-
00054-54-209
10 114 PaS
-12 -9 -6 -3 0 3 6 9
4.11
4.12
4.13
4.14
4.15
4.16
4.17
111
200
220
311
222
400
331
420
422
333
La
ttic
e p
ara
me
ter
(Å)
½s2
hkl sin
2 + 2s
1
hkl (TPa
-1)
0aslope =
a0
a0 = (4.1394 0.0006) Å
= -(1.22 0.06) GPa
http://www.abinit.org.J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 80/4 (1998) 891.H. Yao, L. Ouyang, W.-Y. Ching, J. Am. Ceram. Soc. 90/10 (2007) 3194.
D. Rafaja, C. Wüstefeld, J. Kutzner, A.P. Ehiasarian, M. Šíma, V. Klemm, D. Heger, J. Kortus, Z. Krist. 225 (2010) 599-609.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Stress-free lattice parameter of a Mo thin film
17
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.3135
0.3140
0.3145
0.3150
0.3155
0.3160110
200
220
310
222
321
(a)
211
La
ttic
e p
ara
me
ter
(nm
)
f() = ½s2
hkl sin
2 + 2s
1
hkl (10
-12Pa
-1)
0.0 0.2 0.4 0.6 0.8
0.3140
0.3145
0.3150
0.3155
0.3160110
200211
220
310
222
321
(b)
La
ttic
e p
ara
me
ter
(nm
)
sin2
D. Rafaja, H. Köstenbauer, U. Mühle, C. Löffler, G. Schreiber, M. Kathrein,
J. Winkler, Thin Solid Films 528 (2013) 42-48.
A.G. Every, A.K. McCurdy, in D.F. Nelson (ed.): Springer Materials – The
Landolt-Börnstein Database (http://www.springermaterials.com);
http://dx.doi.org/10.1007/10046537_11.
A. Reuss, Z. angew. Math. Mech. 9 (1929) 49.
A.J. Perry, V. Valvoda, D. Rafaja, Thin Solid Films 214 (1992) 169-174.
001
2
221
0
4421
12110
01211221
0121
12sin
;3;
afassaa
SSSS
SSSsSSs
hkhkhk
hkhk
X-ray elastic constants of Mo:
S11 = 2.63 (1/TPa), S12 = −0.68 (1/TPa),
S44 = 9.20 (1/TPa)
Reuss approach
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Stress-free lattice parameter of a Mo thin film
18
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.3144
0.3145
0.3146
0.3147
0.3148
0.3149
0.3150 DC/25°C
RF/25°C
DC/350°C
DC/150°C
RF/150°C
Substitutional atoms (at.%)
Lattic
e p
ara
mete
r (n
m)
Intrinsic value
𝑎 Mo, Sub = 0.314817 nm −1.099 × 10−4[ nm at.%] × 𝑐 Sub [at.%]
𝑎 Mo, Sub = 0.315017 nm −1.099 × 10−4[ nm at.%] × 𝑐 Sub [at.%]
Substitutional atoms: Fe and Cr
Effect of
interstitial atoms
D. Rafaja, H. Köstenbauer, U. Mühle, C. Löffler, G. Schreiber, M. Kathrein, J. Winkler, Thin Solid Films 528 (2013) 42-48.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Vegard-like dependence of the stress-free lattice parameter
19
D. Rafaja, C. Wüstefeld, M. Dopita, M. Růžička, V. Klemm, G. Schreiber,
D. Heger, M. Šíma, Surf. Coat. Technol. 201 (2007) 9476-9484.
D. Rafaja, C. Wüstefeld, M. Dopita, V. Klemm, D. Heger, G. Schreiber,
M. Šíma, Surf. Coat. Technol. 203 (2008) 572-578.
D. Rafaja, C. Wüstefeld, M. Motylenko, C. Schimpf, T. Barsukova, M.R.
Schwarz, E. Kroke, Chem. Soc. Rev. 41 (2012) 5081-5101.
0.0 0.2 0.4 0.60.405
0.410
0.415
0.420
0.425
0.445
0.450
0.455
0.460
La
ttic
e p
ara
me
ter
(nm
)
mol AlN in TM1-x
AlxN
Zr1-xAlxN: a = [0.458(1) – 0.026(1)·x] nm
Ti1-xAlxN: a = [0.42418(2) – 0.01432(2)·x] nm
Cr1-xAlxN: a = [0.41486(2) – 0.00827(1)·x] nm
Cr1-xAlxN
Ti1-xAlxN
Zr1-xAlxN
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Phase diagrams of the Ti1-xAlxN
20
B. Alling, A.V. Ruban, A. Karimi, O.E. Peil, S.I. Simak,
L. Hultman, I. A. Abrikosov, Phys. Rev. B 75 (2007) 045123.
P.H. Mayrhofer, D. Music, J.M. Schneider, Appl. Phys. Letters
88 (2006) 071922.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Thermodynamically metastable system stabilized by low mobility of adatoms
21
Lattice parameters of (TM,Al)N with TM = Cr, Ti, Zr
Latt
ice p
ara
mete
r
TMN(TM,Al)N
Latt
ice p
ara
mete
r
[TM]/([TM]+[Al])
Vegard-like dependence of the lattice parameters
Combination of the chemical analysis and the structure analysis
Lattice parameter of TM1-xAlxN increases with increasing TM content
Single phase
Dual phase(TM,Al)N + AlN
Overall
composition
Composition
of fcc-(TM,Al)N
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Coexistence of fcc-(Ti,Al)N and w-AlN
22
20 30 40 50 60 70
111
200
220
311
222
400
331
420
422
511/3
33
100
002
101
102
110
103
112
201
202
203
114
212
213
Inte
nsity (
arb
.units)
Diffraction angle (°2)
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As seen by XRD:
Minor phase consisting of
small crystallites is mainly
visible via ‘diffuse scattering’
The stress-free lattice
parameter corresponds to
the chemical composition of
the respective phase
D. Rafaja, Ch. Wüstefeld, M. Dopita, M. Motylenko,
C. Baehtz, IUCrJ 1 (6) (2014) 446-456.
Phase transition fcc w
23
2 nm
__[120]
[001]
[112]
_[111]
w fcc
D. Rafaja, C. Wüstefeld, M. Dopita, M. Motylenko, C. Baehtz, C. Michotte, M. Kathrein, Surf. Coat. Technol. 257 (2014) 26-37.
_[110]
_[210]
Stacking faults
24
(11-20)_
(101)
111 0001
(111)
(0002)
Fm3m P63mc
fcc
wurtziteA
bB
c
C
a
A
bB
c
C
aA
A
bB
aA
bB
aA
bB
aA
Phase transition via formation of stacking faults on the (111)fcc and (0002)w planes
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
D. Rafaja, C. Wüstefeld, M. Dopita, M. Motylenko, C. Baehtz, C. Michotte, M. Kathrein, Surf. Coat. Technol. 257 (2014) 26-37.
Correlated positions of stacking faults
25
Layer Microstructure feature/defect Sequence of the lattice planes
1 Regular fcc stacking ABCABC
2 Intrinsic SF ABCAB|ABC
3 Alternating arrangement of SFs (narrowly spaced SFs) ABCAB|AB|ABC
4 Twin ABC|B|A|C
5 Extrinsic SF ABCA|C|BCABC
S. Martin, C. Ullrich, D. Šimek, U. Martin, D. Rafaja, J. Appl. Cryst. 44 (2011) 779-787.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Stacking along (111)fcc || (002)hcp
26D. Borisova, V. Klemm, S. Martin, S. Wolf, D. Rafaja, Adv. Eng. Mat. 15 (2013) 571-582.
... as seen by HRTEM & FFT/HRTEM
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Microstructure of deformed TRIP steels
27
... as seen by EBSD (phase contrast & band contrast)
fcc austenite
band contrast
SG: Fm-3m
a = 3.5953 Å
hcp -martensite
SG: P63/mmc
a = 2.540 Å
c = 4.111 Å
bcc ’-martensite
SG: Im-3m
a = 2.875 Å
hcpfcc
hcpfcc
dd
aa
002111
2
S. Martin, C. Ullrich, D. Šimek, U. Martin, D. Rafaja, J. Appl. Cryst. 44 (2011) 779-787.D. Borisova, V. Klemm, S. Martin, S. Wolf, D. Rafaja, Adv. Eng. Mat. 15 (2013) 571-582.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Paterson & Warren approach
28
m
L
a
d
d
d
dd
affected
22
*
4
31
mlkh
L
a
a
d
d affected
2224
3
mlkh
L
G
Gaa
affected
hkl
hkl
222
04
31
M.S. Paterson, J. Appl. Phys. 23 (1952) 805.
B.E. Warren, E.P. Warekois, J. Appl. Phys. 24 (1953) 951.
B.E. Warren, J. Appl. Phys. 32 (1961) 2428.
C.N.J. Wagner: Local atomic arrangements studied by X-ray diffraction, Gordon and Breach, New York, 1966.
B.E. Warren: X-ray Diffraction, Dover Inc., New York, 1969/1990.
Comments and corrections to the Warren approach:
L. Velterop, R. Delhez, Th.H. de Keijser, E.J. Mittemeijer, D. Reefman, J. Appl. Cryst. 33 (2000) 296.
E. Estevez-Rams, M. Leoni, P. Scardi, B. Aragon-Fernandez, H. Fuess, Phil. Mag. 83 (2003) 4045.
E. Estevez-Rams, U. Welzel, A. Pentón Madrigal, E.J. Mittemeijer, Acta Cryst. A64 (2008) 537.
lkhLklKhkH ;;21
21
fcc (hkl) hcp (HKL):
fcc SF hcp
222,311,200
fcc
400,220,111
222,131,022,111
Anisotropy of fcc lattice parameters
29
Measured lattice parameters: Macroscopic (residual) stress (crystal anisotropy of the elastic constant s1(hkl)) Stacking faults with the density (contrast factors of stacking faults) Sample displacement p (if not corrected using internal standard)
Austenite (fcc), symmetrical diffraction geometry, uniaxial stress
cotcos4
3
3
3
cotcos4
3
100
1
111
1
100
1
0
0
100
1
111
1
100
11
1
0
0
pG
sssa
aa
ssss
pGsa
aa
hk
hk
hk
hkl
hk
hkl
hkhk
2222
222222
lkh
lhlkkhhkl
Strong correlation between and G for low-angle diffraction lines
0.0 0.1 0.2 0.3
-0.4
-0.2
0.0
0.2
0.4
0.6
111
200
220
311222
400
331
420422 333511
Ghkl
hkl
mlkh
L
Gaffected
hkl
222
hklhklG 42.247.0
B.E. Warren: X-ray Diffraction, Dover Inc., New York, 1969/1990.
R.W. Vook, F. Witt, J. Appl. Phys. 36 (1965) 2169.D. Šimek, D. Rafaja, M. Motylenko, V. Klemm, G. Schreiber, A. Brethfeld, G. Lehmann, steel research int. 79 (10) (2008) 800.D. Rafaja, C. Krbetschek, D. Borisova, G. Schreiber, V. Klemm, Thin Solid Films 530 (2013) 105.
… due to applied stress and formation of stacking faults
30
X-ray diffraction line broadening
G.K. Williamson, W.H. Hall, Acta Metallugica 1 (1953) 22-31.
The Williamson-Hall plot
Size of the diffraction vector (4 sin/)
XR
D lin
e b
roa
de
nin
g
Reciprocal crystallite size
Slope is proportional
to microstrain
tan2cos
2
sin2
2
12
2
12
D
K
D
Kq
0.4 0.6 0.8 1.04
5
6
7
8
9
110
200
211
220 310
222
321
Lin
e b
roadenin
g (
10
-3 n
m-1)
sin
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Δ 𝑞 212 =
4𝜋
𝜆sin 𝜃
Δ𝑑
𝑑
212
= 𝑞 𝜀212
Anisotropic XRD line broadening
310.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
110 200
211 220
310
222 3
21
(b)
2
hkl (
10
-3n
m-2)
sin2
0.0 0.3 0.6 0.9 1.2 1.50
2
4
6
8
110 200
211 220
310
222 3
21
(a)
2
hkl (
10
-3n
m-2)
(1-)*sin2
𝜀ℎ𝑘𝑙2 = 2𝜋𝑏2𝑀2𝐶ℎ𝑘𝑙𝜚𝑑𝑖𝑠𝑙
𝐶ℎ𝑘𝑙 = 𝐶100 1 − 𝜁Γ
Γ =ℎ2𝑘2 + 𝑘2𝑙2 + 𝑙2ℎ2
ℎ2 + 𝑘2 + 𝑙2 2
𝛽ℎ𝑘𝑙2 =
0.9
𝐷
2
+ 4 𝜀ℎ𝑘𝑙2 sin 𝜃
𝜆
2
𝛽ℎ𝑘𝑙2 =
0.9
𝐷
2
+
+8𝜋𝑏2𝑀2𝐶100
𝜆𝜚𝑑𝑖𝑠𝑙 1 − 𝜁Γ sin 𝜃 2
𝛽ℎ𝑘𝑙2 =
0.9
𝐷
2
+ 4 𝜀1002 1 − 𝜁Γ
sin 𝜃
𝜆
2
Electrical conductivity of Mo thin films
32
Effect of individual microstructure defects
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
𝜚 = 𝜀1002 × 1.10 ± 0.01 × 105µΩcm + Δ𝑎 × 7.47 ± 0.04 × 103 µΩcm nm
+𝑐𝑠𝑢𝑏𝑠𝑡 × 4.18 ± 0.26 µΩcm at.% + 7.87 ± 0.27 µΩcm
Intrinsic electrical conductivity of molybdenum: 5.46 µΩcm
Grain boundaries: (2.41±0.09) µΩcm
Dislocations: 𝜀1002 × 1.10 ± 0.01 ×
105µΩcm;
max. (3.63±0.04) µΩcm
Substitutional atoms: 𝑐𝑠𝑢𝑏𝑠𝑡 ×4.2 ± 0.3 µΩcm at.%;
max. 11.7 µΩcm @ 2.8 at.%
Interstitial atoms: 𝑐𝑖𝑛𝑡𝑒𝑟 × 4.5 ±0 50 100 150 200 250 300 350
5
10
15
20
25
Resis
tivity (c
m)
Temperature (°C)
D. Rafaja, H. Köstenbauer, U. Mühle, C. Löffler, G. Schreiber, M. Kathrein,
J. Winkler, Thin Solid Films 528 (2013) 42-48.
Anisotropic line broadening (hexagonal)
𝛽ℎ𝑘𝑙 =2𝑙
𝜋𝑐2𝑑ℎ𝑘𝑙
𝛾
1 − 𝛾
C. Schimpf. M. Motylenko, D. Rafaja, Mater. Char. 86 (2013) 190
Turbostratic disorder (𝛾 = 0.07)
Basal plane corrugations (∆𝑐 = 0.05 Å)
Dislocations E001<110> (𝑀2𝜚 = 5 × 1013 m−2)
Basal plane stacking faults (𝛼𝑠𝑓 = 0.015)
𝛽ℎ𝑘𝑙 =𝑙2
𝑐3𝑑ℎ𝑘𝑙 ∆𝑐
𝛽ℎ𝑘𝑙 = 𝜒ℎ𝑘𝑙
1
𝑑ℎ𝑘𝑙𝑏
𝜋
2𝑀2𝜚
𝛽ℎ𝑘𝑙 =3𝑙
𝑐2𝑑ℎ𝑘𝑙𝛼𝑠𝑓
A.V. Kurdyumov, Sov. Phys. Cryst. 20 (1976) 596 | K. Ufer et al., Z. Krist. 219 (2004) 519
P. Klimanek & R. Kužel. J. Appl. Cryst. 21 (1988) 59 | R. Kužel & P. Klimanek, ibid. 363
B.E. Warren, X-ray diffraction, Addison-Wesley, Mass., USA (1969)
33
34
X-ray diffraction line broadening
Size of the diffraction vector
XR
D lin
e b
roadenin
gReciprocal
cluster size
Reciprocal crystallite size
Misorientation
of adjacent
crystallites
G.K. Williamson, W.H. Hall, Acta Metallugica 1 (1953)
22-31.
D. Rafaja, V. Klemm, G. Schreiber, M. Knapp, R. Kužel,
J. Appl. Cryst. 37 (2004) 613-620.
The Williamson-Hall plotModified Williamson-Hall
plot
Size of the diffraction vector (4 sin/)
XR
D lin
e b
roadenin
g
Reciprocal crystallite size
Slope is proportional
to microstrain
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
tan2cos
2
sin2
2
12
2
12
D
K
D
Kq
35
Phenomenon of the partial coherence of crystallites
222
33
*
*
hklhklhklhkl
rqi
V
rqi
FFTFFTqI
qFFTrderrrderqF
Diffraction on a single crystallite:
1
1 11
2
11
2
1
cos2K
j
jK
i
jiijii
K
i
i
K
i
i
K
i
i
K
i
i
RRqqFqFqFqI
qFqFqFqI
Diffraction on a cluster of mutually (partially)
coherent crystallites:
D. Rafaja, V. Klemm, G. Schreiber, M. Knapp, R. Kužel, J. Appl. Cryst. 37 (2004) 613.
Kinematical
approximation
Coherence term: interference of waves diffracted
by different crystallites
36
Phenomenon of the partial coherence of crystallites
222
33
*
*
hklhklhklhkl
rqi
V
rqi
FFTFFTqI
qFFTrderrrderqF
Diffraction on a single crystallite:
1
1 11
2
11
2
1
cos2K
j
jK
i
jiijii
K
i
i
K
i
i
K
i
i
K
i
i
RRqqFqFqFqI
qFqFqFqI
D. Rafaja, V. Klemm, G. Schreiber, M. Knapp, R. Kužel, J. Appl. Cryst. 37 (2004) 613.
-10 -5 0 5 10
0
2
4
6
8
10
Y A
xis
Titl
e
X axis title
Intensity
q
Kinematical
approximation
Coherence term: interference of waves diffracted
by different crystallites
Diffraction on a cluster of mutually (partially)
coherent crystallites:
Simulation of the reciprocal space maps
37
000
111
200
002202
311
004
113313
313
004
113
311
204
204
202
115115
315
315
200
_111
_113
_113
_204_
204
_115
_115
_202
_202
_200
_200
_311
_311
_313
_313
_315
_315
qx
qz
Mutually rotated and
shifted spherical
crystallites
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Intensity = |F1|2 + |F
2|2
qx [Å-1]
q
z [
Å-1
]
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Intensity = |F1|2 + |F
2|2 + |F
1||F
2| cos(q.D)
qx [Å-1]
q
z [
Å-1
]
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
14x 10
11
qz [Å-1]
Inte
nsity (
arb
.units)
q = 0.03
1
1 11
2cos2
K
j
jK
i
jiijii
K
i
i RRqqFqFqFqI
D. Rafaja, V. Klemm, Ch. Wüstefeld, M. Motylenko,
M. Dopita, M. Schwarz, T. Barsukova, E. Kroke,
Z. Kristallogr. Suppl. 27 (2008) 15.
Simulation of the reciprocal space maps
38
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Intensity = |F1|2 + |F
2|2
qx [Å-1]
q
z [
Å-1
]
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Intensity = |F1|2 + |F
2|2 + |F
1||F
2| cos(q.D)
qx [Å-1]
q
z [
Å-1
]
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
2
4
6
8
10x 10
11
qz [Å-1]
Inte
nsity (
arb
.units)
q = 0.14
000
111
200
002202
311
004
113313
313
004
113
311
204
204
202
115115
315
315
200
_111
_113
_113
_204_
204
_115
_115
_202
_202
_200
_200
_311
_311
_313
_313
_315
_315
qx
qz
Mutually rotated and
shifted spherical
crystallites
1
1 11
2cos2
K
j
jK
i
jiijii
K
i
i RRqqFqFqFqI
D. Rafaja, V. Klemm, Ch. Wüstefeld, M. Motylenko,
M. Dopita, M. Schwarz, T. Barsukova, E. Kroke,
Z. Kristallogr. Suppl. 27 (2008) 15.
Simulation of the reciprocal space maps
39
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Intensity = |F1|2 + |F
2|2
qx [Å-1]
q
z [
Å-1
]
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Intensity = |F1|2 + |F
2|2 + |F
1||F
2| cos(q.D)
qx [Å-1]
q
z [
Å-1
]
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250
1
2
3
4
5
6x 10
11
qz [Å-1]
Inte
nsity (
arb
.units)
q = 0.4
000
111
200
002202
311
004
113313
313
004
113
311
204
204
202
115115
315
315
200
_111
_113
_113
_204_
204
_115
_115
_202
_202
_200
_200
_311
_311
_313
_313
_315
_315
qx
qz
Mutually rotated and
shifted spherical
crystallites
1
1 11
2cos2
K
j
jK
i
jiijii
K
i
i RRqqFqFqFqI
D. Rafaja, V. Klemm, Ch. Wüstefeld, M. Motylenko,
M. Dopita, M. Schwarz, T. Barsukova, E. Kroke,
Z. Kristallogr. Suppl. 27 (2008) 15.
Effect of the partial coherence on the XRD line broadening
Diffraction vector
XR
D l
ine b
road
en
ing
000
111
200
002 202
311
004
113313
313
004
113
311
204
204
202
115115
315
315
200
_111
_113
_113
_204_
204
_115
_115
_202
_202
_200
_200
_311
_311
_313
_313
_315
_315
qx
qz
Non-overlapping reciprocal lattice points: “classical” kinematical diffraction theory
Size of the reciprocal
lattice points
0cos1
1 1
K
j
jK
i
jiijii RRqqFqF
K
i
i qF1
2
40
44141
Effect of the partial coherence on the XRD line broadening
Diffraction vector
XR
D l
ine b
road
en
ing
000
111
200
002 202
311
004
113313
313
004
113
311
204
204
202
115115
315
315
200
_111
_113
_113
_204_
204
_115
_115
_202
_202
_200
_200
_311
_311
_313
_313
_315
_315
qx
qz
Partially overlapping reciprocal lattice points: mutual coherence of crystallites
0cos1
1 1
K
j
jK
i
jiijii RRqqFqF
Size of the reciprocal
lattice points
42
XRD line broadening and microstructure features
Diffraction vector
XR
D l
ine b
road
en
ing
Reciprocal
crystallite size
Non-overlapping reciprocal lattice points: XRD distinguishes individual crystallites
0cos1
1 1
K
j
jK
i
jiijii RRqqFqF
Scherrer formula is valid
K
i
i qF1
2
43
XRD line broadening and microstructure features
qx
qz
Disorientation
of crystallites
Diffraction vector
XR
D l
ine b
road
en
ing
Disorientation
of crystallites
Onset of the partial coherence: XRD sees the disorientation of individual crystallites
Reciprocal
crystallite size
44
XRD line broadening and microstructure features
Partial overlap of the reciprocal lattice points: XRD sees the crystallites larger
Diffraction vector
XR
D l
ine b
road
en
ing
Disorientation
of crystallites
Reciprocal
crystallite size
0cos1
1 1
K
j
jK
i
jiijii RRqqFqF
45
XRD line broadening and microstructure features
Extrapolation to q = 0: in analogy with the “classical” diffraction theory, XRD
sees the size of the clusters of crystallites
Diffraction vector
XR
D l
ine b
road
en
ing
Size of clusters of
partially coherent
crystallites
Disorientation
of crystallites
Reciprocal
crystallite size
Elastic anisotropy & partial coherence
46
0.2 0.3 0.4 0.5 0.65
10
15
20
25
30
35
111
200
220
311
222
400
331
420
422
511
Lin
e b
roa
den
ing
(1
0-3 Å
-1)
sin
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
47
(Cr,Al,Si)N nanocomposites
CrystallitesClusters
Cr0.92Al0.08N0.2 0.4 0.6 0.8 1.00.02
0.04
0.06
0.08
0.10
0.12
Lin
e b
roadenin
g (
nm
-1)
sin
Crystallite size: 9 nm
Cluster size: 36 nm
=0.50°=0.56°
5 nm
Crystallites
Clusters
Cr0.92Al0.08N
D. Rafaja, Ch. Wüstefeld, M. Dopita, M. Růžička, V. Klemm, G. Schreiber, D. Heger, M. Šíma, Surf. Coat. Technol. 201 (2007) 9476.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
(Ti,Al,Si)N nanocomposites
48
20 40 60 80 1000.1
0.2
0.3
0.4
0.5
Lin
e b
roadenin
g [nm
-1]
q [nm-1]
… Ti0.38Al0.62N
… Ti0.40Al0.53Si0.07N
Crystallite size
D = (2.4 0.2) nm
Cluster size
D = (8.5 1.0) nm
Partially coherent crystallites
Non-coherent crystallites
2 nm
Ti0.38Al0.62N – partially coherent
2 nm
Ti0.40Al0.53Si0.07N – non coherent
D. Rafaja, A. Poklad, V. Klemm, G. Schreiber, D. Heger, M. Šíma, Mat. Sci. Eng. A 462 (2007) 279.
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
49
“Local heteroepitaxy” of crystallites as seen by TEM
2 nm
fcc-(Ti,Al)N and w-AlN nanocrystallites are
separated by a-Si3N4
Local heteroepitaxy between fcc-(Ti,Al)N
and w-AlN nanocrystallites
Crystallite size: (2.4 0.2) nm
Cluster size: (8.5 1.0) nmTi0.40Al0.53Si0.08N
Crystallite size: (2.4
0.2) nm
2 nmTi0.38Al0.62N
CrystallitesClusters of crystallites
Cubic
phase
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
50
“Local heteroepitaxy” of crystallites as seen by XRD
20 40 60 80 1000.1
0.2
0.3
0.4
0.5
Lin
e b
roadenin
g [nm
-1]
q [nm-1]
… Ti0.38Al0.62N
… Ti0.40Al0.53Si0.08N
Crystallite size
(2.4 0.2) nm
Cluster size
(8.5 1.0) nm
D. Rafaja, A. Poklad, V. Klemm, G. Schreiber, D. Heger, M. Šíma, Freiberger Forschungshefte B 331 (2005) 36-38.
Partially coherent crystallites
Non-coherent crystallites
0.0 0.2 0.4 0.6 0.80.410
0.413
0.416
0.419
0.422
0.425
Latt
ice
pa
ram
ete
r [n
m]
sin2
|| << 1 GPa
H = (25.4 0.6) GPa
Local
heteroepitaxy
= -(9.0 0.5) GPa
H = (30.6 1.3) GPa
No local heteroepitaxy
The sin² plot Modified Williamson-Hall
plot
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
51
Effect of “local heteroepitaxy” on the hardness
2 3 4 5 6 7 8 924
27
30
33
36
1
2
3
4
5
1
2
3
4
5
Ha
rdn
ess (
GP
a)
Crystallite size (nm)
2 3 4 5 6 7 8 924
27
30
33
36
D. Rafaja, A. Poklad, V. Klemm, G. Schreiber, D. Heger, M. Šíma, Mat. Sci. Eng. A 462 (2007) 279–282.
“Hall-Petch“ and inverse
“Hall-Petch” relationship
describing the dependence
of the hardness on the
crystallite size
Considerable contribution
of the local heteroepitaxy
to the hardness
… Ti1-xAlxN
… Ti1-x-yAlxSiyN
Partially
coherent
crystallites
Non-coherent
crystallites
FRIENDS2, Advanced coating and characterization techniques, Microstructure studies
Summary
52
X-ray diffraction
Non-destructive method
Analysis of large sample volumes
Visualization of microstructure defects is possible only through the strain fields
Excellent precision of the lattice spacing determination
Transmission electron microscopy
Destructive method
Local analytical method
Direct visualization of microstructure defects is possible
Medium precision of the lattice spacing determination
53
Acknowledgement
Materials Science (TU Freiberg)
Ch. Wüstefeld
Dr. M. Motylenko
A. Poklad
Ch. Schimpf
Dr. V. Klemm
Dr. U. Mühle
Dr. M. Dopita
R. Popp
A. Leuteritz
G. Schreiber
Dr. D. Heger
D. Chmelik
U. Ratayski
Inorganic Chemistry(TU Freiberg)
Prof. Dr. E. Kroke
Dr. M.R. Schwarz
T. Barsukova
Helmholtz Centre(Dresden)
ESRF (Grenoble)
Dr. C. Bähtz
TU Darmstadt
HASYLAB Hamburg
Dr. M. Knapp (KIT)CERATIZIT
C. Michotte (Mamer)
Dr. M. Kathrein & Ch. Czettl (Reutte)
SHM Šumperk
Dr. M. Šíma
Dr. M. Jílek
University of Bayreuth
Dr. D. Frost
Dr. L. Dubrovinsky
Prof. N. Dubrovinskaia
Plansee CM (Lechbruck)
Dr. P. Polcik
C. Polzer
Helmholtz Centre(Potsdam)
HASYLAB Hamburg
Dr. Ch. Lathe
Theoretical Physics(TU Freiberg)
Prof. Dr. J. Kortus
J. Kutzner
Sheffield Hallam University
Prof. Dr. A.P. Ehiasarian
Plansee SE (Reutte)
Dr. H. Köstenbauer
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Financial support
Thank you for your kind attention
Supported from the funds of the European Community and the Saxony Government
Dr. Erich Krüger
Research Foundation:
Freiberg High-Pressure
Research Centre