Combined x-ray Texture-Structure-Microstructure analysis applied to
ferroelectric ultrastructures: a case study on Pb0.76Ca0.24TiO3
L. Cont ab, D. Chateigner a, L. Lutterotti b, J. Ricote c, M.L. Calzada c, J. Mendiola c
LPEC, Univ. Le Mans, France DIM, Univ. di Trento, Italia DMF-ICMM-CSIC, Cantoblanco, Madrid, España
Summary • Usual up-to-date approaches for polycrystals
– Texture – Structure-Microstructure – Problems on ultrastructures
• Combined approach – Experimental needs – Methodology-Algorithm – Ultrastructure implementation – Case study on Pb0.76Ca0.24TiO3
• Future trends
Usual Texture Analysis X=010
Y=100Z=001
cα
β
X
Y
φχχχφπ
χφ d d nsi )P( 41 =
V)dV(
{hkl} pole figure measurement + corrections:
dg (g)f 81 =
VdV(g)
2π
We want f(g) (ODF): with g = (α,β,γ) X=010
Y=100Z=001
c
a
b
αβ
γX
Y
∫><
ϕπ y//hkl
hkl~d f(g)
21 = )y(P
!
!
We have to invert (Fundamental equation of Texture Analysis):
( ) ⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
∏+
hkl
I1
nhkl
0n1n
)y(P
)g(f )g(f N )g(f!
WIMV refinement method: Williams-Imhof-Matthies-Vinel
Usual Structure-Microstructure Analysis
)bkg(2)2(S)2(I = )I(2phases hkl,
phases hkl,phases hkl, θ+θθθ ∑
(Full pattern fitting, Rietveld Analysis) Si3N4 matrix with SiC whiskers:
Random powder:
2c
Phkl
2hklhkl V
LmFS)2(I =θ
S: scale factor (phase abundance) Fhkl: structure factor (includes Debye-Waller term) Vc: unit-cell volume
)2(S*)2(S)2(S Shkl
Ihklhkl θθ=θ
SI: instrumental broadening SS: Sample aberrations
crystallite sizes (iso. or anisotropic) rms microstrains ε
Problems on ultrastructures
Ferroelectric film (PTC) Electrode (Pt)
Antidiffusion barrier (TiO2) Oxide (native, thermally grown)
SC Substrate (Si)
- Strong intra- and inter-phase overlaps - Mixture of very strong and lower textures - texture effect not fully removable: structure - structure unknown: texture
001/
100
PTC
111
PTC
+ 1
11 P
t
011/
110
PTC
+ S
i (λ/
2)
Sum diagram
Need something !!
Combined approach Experimental needs
ω = 20° ω = 40°
Mapping Spectrometer space for correction of: - instrumental resolution - instrumental misalignments
χ
60°
0°
χ
60°
0°
Methodology-Algorithm
Rietveld Structure, Microstructure
WIMV Texture
Rietveld and WIMV algorithm are alternatively used to correct for each others contributions: Marquardt non-linear least squares fit is used for the Rietveld.
Pole figure extraction (Le Bail method): Phkl(χ,ϕ)
Correction of intensities for texture:
Ihkl(2θ,χ,ϕ) = Ihkl(2θ) Phkl(χ,ϕ)
Ultrastructure implementation
Corrections are needed for volumic/absorption changes when the samples are rotated. With a CPS detector, these correction factors are:
( )( ) ( )( )χωµ−−χµ−−=χ cossin/T2exp1/cos/Tgexp1gC 21film top
( )( ) ( )( )χωµ−χµ−= ∑∑χχ cossin/T2exp/cos/TgexpCC 'i
'i
'i
'i2
film toplayer cov.
Gives access to individual Thicknesses in the refinement
PTC
Pt
a = 3.945(1) Å c = 4.080(1) Å T = 4080(10) Å tiso = 390(7) Å ε = 0.0067(1)
a = 3.955(1) Å T' = 462(4) Å t'iso = 458(3) Å ε' = 0.0032(1)
Case study on Pb0.76Ca0.24TiO3
WIMV vs Entropy modified WIMV approach
WIMV
E-WIMV
Better refinement with E-WIMV: - lower reliability factors (structure and texture) - better high density level reproduction
Texture PtTextureIndex(m.r.d.2)
PTCTextureIndex(m.r.d.2)
PtRP0
(%)
PTCRP0
(%)
Rw
(%)
RBragg
(%)WIMV 48.1 1.3 18.4 11.4 12.4 7.7EWIMV 40.8 2 13.7 11.2 7 4.7
Future trends - Combining with reflectivity measurements: independently measured and refined thicknesses, electron densities and roughnesses - Adding residual stress determinations - Multiple Analysis Using Diffraction, a web-based tutorial for the combined approach: search MAUD (Luca Lutterotti) - Quantitative Texture Analysis Internet Course: http://lpec.univ-lemans.fr/qta (Daniel Chateigner)
Acknowledgements This work is funded by EU project ESQUI (http://lpec.univ-lemans.fr/esqui : an x-ray Expert System for microelectronic film QUality Improvement, G6RD-CT99-00169), under the RTD program.
Quantitative Texture Analysis Internet Course
http://lpec.univ-lemans.fr/qta
