Röntgenbeugung und Röntgenstreuung an
Multilagenschichten mit diskontinuierlichen Grenzflächen
David Rafaja
Institut für Metallkunde
Struktur und Gefüge von Werkstoffen
TU Bergakademie Freiberghttp://www.ww.tu-freiberg.de/mk/
2 Physikalisches Kolloquium TUC, 18.6. 2003
Outlines
Structure model of multilayers with non-continuous interfaces
Experimental methods and theoretical approaches for structure investigation of multilayers
– X-ray reflectivity (XRR)
– Small-angle X-ray scattering
– Reciprocal space mapping
– Wide-angle XRD
Applications, Examples (Fe/Au)
3 Physikalisches Kolloquium TUC, 18.6. 2003
Microstructure Model
Fe/Au-Multilayer
TEM courtesy of Prof. J. Zweck, University of Regensburg
Anticipated changes of the multilayer microstructure
(after a temperature treatment)
10 nm
4 Physikalisches Kolloquium TUC, 18.6. 2003
Mean thickness of individual layers in the periodic motif
Mean interface roughness Mean interplanar spacing
(residual stresses) Mean intralayer and interlayer
disorder (atomic ordering) Crystallite size and texture
Interface continuity
Electron density of individual layers
Thickness of individual layers Interface roughness Interface morphology
(geometrical and diffuse roughness, lateral correlation length)
Replication of the interference roughness (vertical correlation length)
Interface continuity
From SAXS (small-angle X-ray scattering)
Real Structure of Multilayers
From WAXS (wide-angle X-ray scattering)
5 Physikalisches Kolloquium TUC, 18.6. 2003
Experimental set-up
Scintillationdetector
Flat monochromator
SampleGoebel mirror
X-ray source
Sample rotation,
Normal direction
Diffraction vector
Diffraction angle, 2
Angle of incidence,
Sample inclination,
Used for XRR, SAXS, GAXRD and symmetrical XRD
Can the interface discontinuities be seen by X-
rays?
7 Physikalisches Kolloquium TUC, 18.6. 2003
X-Ray ReflectivityTheoretical background
Substrate
- Multiple (dynamical) scattering of X-rays
- Optical theory for smooth interfaces (no interface roughness)
22
1
11,
212
1,
21,
11
1
cos4
sin4
2exp;
1
jjj
jjjjj
jjjj
jjjjj
jjjjj
j
Rj
nq
tiqfqq
qqr
fAfr
AfrA
E
E
Based on: L.G. Parrat, Phys. Rev. 95 (1954) 359.
Recursive formula
2NRI
12
1 0
2
jjjatj
ej fiffN
rn
8 Physikalisches Kolloquium TUC, 18.6. 2003
X-Ray ReflectivityTheoretical background
Substrate
22
21,
1
11,
212
1,
21,
11
1
cos4
;2exp
2exp
1
jjjjj
jjjjj
jjjj
jjjjj
jjjjj
j
Rj
nqtiqf
qqq
qqr
fAfr
AfrA
E
E
Change in the Fresnel reflection coefficient (Debye-Waller factor)
X-ray reflectivity of multilayers with a certain interface roughness
L. Névot, P. Croce, Rev. Phys. Appl. 15 (1980) 761.
G.H. Vineyard, Phys. Rev. B 26 (1982) 4146.S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys. Rev. B 38 (1988) 2297.
DWBA
The interfaces must be continuous
tj
tj-1
j-2
9 Physikalisches Kolloquium TUC, 18.6. 2003
X-ray Diffuse Scatteringon continuous interfaces
S
yqxqiyxCq
jjjjj
jjj
yxz eedxdyF
kkkk
kt
1~
8exp2
,
221
11,
2
Distorted wave Born approximation - DWBA
Differential cross-section of the diffuse scattering
S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys. Rev. B 38 (1988) 2297.V. Holý, J. Kuběna, I. Ohlídal, K. Lischka, W. Plotz, Phys. Rev. B 47 (1993) 15896.V.Holý, T.Baumbach, Phys. Rev. B 49 (1994) 10668.
Substrate
C (x,y) … In-plane correlation of interface corrugations
In a multilayer: additionally the vertical correlation
10 Physikalisches Kolloquium TUC, 18.6. 2003
X-ray Reflectivity
Structure model
Substrate
Layer A
Layer B
Layer C
Layer X
Capping layer
J.H. Underwood, T.W. Barbee, Appl. Opt. 20 (1981) 3027.P. Lee, Appl. Opt. 22 (1983) 1241.B. Vidal, P. Vincent, Appl. Opt. 23 (1984) 1794.S.K. Sinha, E.B. Sirota, S. Garoff, H.B. Stanley, Phys. Rev. B 38 (1988) 2297.V. Holý, J. Kuběna, I. Ohlídal, K. Lischka, W. Plotz, Phys. Rev. B 47 (1993) 15896.
0;0 zyx qqq
z
, t, (top)
, t, (X)
, t, (C)
, t, (B)
, t, (A)
, (S)
11 Physikalisches Kolloquium TUC, 18.6. 2003
XRR Curve of a Periodic Multilayer
0 2 4 6 8 1010
-8
10-6
10-4
10-2
100
102
104
106
108
Au/Al, 10x, tA+t
B=7.5nm
t(A)/t(B)=1/1
t(A)/t(B)=1/2
t(A)/t(B)=1/3
t(A)/t(B)=1/4
Re
fle
cti
vit
y
Glancing angle (o2)
Total reflection Electron density of the uppermost layer
Decrease of the reflected Intensity interface roughness
Kiessig oscillations thickness of the whole multilayer
Bragg-like peaks thickness of the periodic motif
Extinction of the Bragg-like peaks thickness of the individual layers in the multilayer system
12
1 0
2 fiff
rn e
e
2exp 224 qqI
mntqt 2cos224
mn 22 cos2
01 AB ttnI
12 Physikalisches Kolloquium TUC, 18.6. 2003
X-ray Diffuse Scattering of a Periodic Multilayer
Reciprocal space mapping Observed phenomena
Yoneda Peaks Maximum of Fresnel transmissions coefficients, t (kin) or t (kout)
Y.Yoneda, Phys. Rev 131 (1963) 2010.
Maximum of resonant diffuse scattering (RDS, Holy‘s bananas) kinematical effect (periodicity of the multilayer)
Bragg-like lines dynamical effect (vertical correlation of corrugations)
Crossing of the RSD and Bragg-like lines V.Holý, T.Baumbach, Phys. Rev. B 49 (1994)
10668.
Information on the mesoscopic Structure in the lateral direction and on the vertical correlation of disturbances
Sample inclination (arcsec)
(a
rcse
c)
-6000 -4000 -2000 0 2000 4000 6000
4000
6000
8000
qx-qz scan at qy = 0
Coplanar diffraction geometry
13 Physikalisches Kolloquium TUC, 18.6. 2003
Fe/Au MultilayersExperimental example
1 2 3 4 5 6 7100
101
102
103
104
105
106
107
108
Inte
nsi
ty (
a.u
.)
Glancing angle (o2)
t (Fe) (27 ± 2) Å
t (Au) (23 ± 1) Å
50 Å
(Fe) 5 Å
(Au) 5 Å
(Fe) (1.4 ± 0.2)
(Au) (0.9 ± 0.1)
Fe/Au (27Å/23Å)x10
Si/Au(100Å)
Refined parameters
14 Physikalisches Kolloquium TUC, 18.6. 2003
Binary System Fe – AuA ssessed A u-F e phase diagram. C alculated.
Au Fe
15 Physikalisches Kolloquium TUC, 18.6. 2003
XRR on Multilayers with Non-Continuous Interfaces
Continuous Discontinuous
Regions
ContinuousDiscontinuous
12
1 jjj AfA
2NAR
Reflectivity
Amplitude and Phase shift
Interfaces
16 Physikalisches Kolloquium TUC, 18.6. 2003
XRR on Multilayers with Non-Continuous Interfaces
Intensity of Bragg peaks decreases
The fringes near the TER are shifted
The structure refinement using the classical model yields closer electron densities of the alternating materials and larger roughness of all interfaces
0 2 4 6 8 1010-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
(c)
(b)
(a)
Ref
lect
ivity
(ar
b.un
its)
Glancing angle (°2)
c = 100%
c = 60%
c = 30%
Changes in the XRR curve
Consequences
Fe/Au (30Å/10Å) x 8
Simulation
17 Physikalisches Kolloquium TUC, 18.6. 2003
Diffuse Scattering from Multilayers with Non-continuous Interfaces
Interfaces
Continuous Discontinuous
S
yqxqiyxCq
jjjjj
jjj
yxz eedxdyF
kkkk
kt
1~
8exp2
,
221
11,
2
kS k
yqxqiyxCq
jjjjjj
yxeedxdyF
ctct
yxz ,1~
1
,
1,1,
2
1
0, yxk 1, yxk Form-factor
DWBA:Differential cross-section
The integration is performed only in the
continuous regions
18 Physikalisches Kolloquium TUC, 18.6. 2003
Diffuse Scattering from Multilayers with Non-continuous Interfaces
D. Rafaja, H. Fuess, D. Šimek, J. Kub, J. Zweck, J. Vacínová, V. Valvoda, J. Phys.: Condensed Matter 14 (2002) 5303-5314.
Consequences
Decrease of the intensity of the Yoneda peaks modified
Fresnel transmission coefficients
Broadening of the specular peak in the longitudinal
scans „convolution“ with the form-factor
-1.0 -0.5 0.0 0.5 1.0100
101
102
103
104
105
106
107
Inte
nsity
(cp
s)
Sample inclination (deg)
19 Physikalisches Kolloquium TUC, 18.6. 2003
0 1 2 3 4 5 6 7100
102
104
106
108
1010
1012
Inte
nsi
ty (
cps)
Diffraction angle (°2)
Diffuse Scattering from Multilayers with Non-continuous Interfaces
Fe/Au (70Å/21Å)13 / 280Å Au / SiO2
As deposited2h/200°C2h/300°C4h/300°C
20 Physikalisches Kolloquium TUC, 18.6. 2003
Wide-Angle X-ray Scattering
Structure model
tA
tB
Continuous and discrete interface
roughness
Intralayer disorder
Average d-spacingInterlayer distance
Jahn-Teller-Method (layered structures) Additional information on the atomic ordering
(interplanar distances, intralayer disorder, texture)
E.E. Fullerton, I.K. Schuller, H. Vanderstraeten and Y. Bruynseraede, Phys. Rev. B 45 (1992) 9292.
21 Physikalisches Kolloquium TUC, 18.6. 2003
Kinematical Theory of WAXS for Multilayers with Continuous
Interfaces
2
2
*
22
1
;
c
aa
caP
eFFFFI
LL
L
atiqLSLSLSL
LL
4
1
1
4exp
:,,
22
1
22
10
1
2
2
qiqd
e
ef
qiqdnfF
ddeePefefF
dnndyxr
L
N
n
N
nLnL
N
n
ddniqiqnd
n
N
n
rqinL
L
LL
L
L
L
Intensity:
Positions of interfaces (Gauss-like distribution):
Positions of individual atoms (correlated displacements):
Structure factor of individual layers:
Interatomic distances and their fluctuations:
22 Physikalisches Kolloquium TUC, 18.6. 2003
WAXS Diffraction Pattern of a Periodic Multilayer
30 32 34 36 38 40 42 44 46 48 500
5
10
15
20
25
30
35
dBd
A
+2
+1-4
-3-2
-1d
0
Inte
nsi
ty (
a.u
.)
Diffraction angle (o2)
n
dn 1sin2
Positions of Satellites:
BABA
BBAA
NNNN
dNdNd
BBAA dNdN
Periodicity of a bi-layer:
Mean interplanar spacing:
Fe/Au (3.24nm/1.41nm) 12Fe: 16 0.20268 nm, Au: 6 0.2355 nm
23 Physikalisches Kolloquium TUC, 18.6. 2003
WAXS on Multilayers with Non-Continuous Interfaces
substrate
buffer
k
Pqik
k
rqiPqi
k
rPqirqi
jP
rqiMP
RqijV
rqiM
jP
rqiM
RqijV
rqiMM
rqiM
jP
rqiP
Rqij
jP
rRqijPP
rqiP
Prqi
PMrqi
MVrqi
kkk
j
j
j
j
efrdeerderde
rdeeRErderEE
rdeeRErderErderE
rdeeRE
rderRErderE
rderErderErderEE
00
000
0
00
000
Structure model
Kinematical Theory
Matrix + Precipitates
24 Physikalisches Kolloquium TUC, 18.6. 2003
WAXS on Multilayers with Non-Continuous Interfaces
MLMMLM
MLM
j
iqtBjAj
iqzML
j
iqtBjAj
iqzM
j k
PqiMP
RqiPqiM
FFccFcFcEI
EEIcFFcEE
eFFeF
eFFecFcEE
effeefEE
Ajj
Ajj
kjjM
Re121
;1
1
222220
0
0
Sample0
,
D. Rafaja, H. Fuess, D. Simek, L. Zdeborova and V. Valvoda, J. Phys.: Condens. Matter 14 (2002) 10021-10032.
f … atomic scattering factors,
F … structure factors,c … continuity of
interfaces,R … positions of
precipitates,E0 … amplitude of the
Thomson scattering,z … origin of the layer
A,t … thickness of the
layer A
Matrix Multilayer Interference Term
25 Physikalisches Kolloquium TUC, 18.6. 2003
WAXS – Simulation of Interface Discontinuity
MLM
ML
M
FFcc
Fc
Fc
EI
Re12
122
22
2
0
32 36 40 44 48
0
200
400
600
800
1000
2
10
-1
-2
-3
-4
(a)
Inte
ns
ity
(a
.u.)
D iffraction angle (o2)
32 36 40 44 48
0
200
400
600
800
1000
(b)
Diffraction angle (o2)
20 % Interface discontinuity 40 %
26 Physikalisches Kolloquium TUC, 18.6. 2003
Combined Refinement SAXS/WAXS
0 2 4 610
0
102
104
106
108
1010
Inte
nsi
ty (
cps)
Scattering angle (°2)
Virgin 2h/200°C XRR XRD XRR XRDt(Fe) 26.5 25.6 26.5 27.0t(Au) 24.0 24.6 22.0 27.8 50.5 50.2 48.5 54.8d(Fe) 2.031 2.027d(Au) 2.359 2.353 0.09 0.13(Fe) 6.5 1.0 7.0 2.0(Au) 6.5 1.2 8.0 2.4(surf) 6.5 9.0c(%) 90 100 85 80
30 35 40 45 5010
2
103
104
Inte
nsi
ty (
arb
.un
its)
Diffraction angle (o2)
Fe/Au (26Å/24Å)10
27 Physikalisches Kolloquium TUC, 18.6. 2003
Fe/Au (26Å/24Å)10Large correlation of the interface roughness
-6000 -4000 -2000 0 2000 4000 60002000
4000
6000
8000
Sample inclination from the normal direction (arcsec)
Diff
ract
ion a
ngle
(a
rcse
c)
Well-pronounced maxima of the resonant diffuse scattering
Large difference between (XRR) and (XRD)
28 Physikalisches Kolloquium TUC, 18.6. 2003
Combined Refinement SAXS/WAXS
0 2 4 610
0
102
104
106
108
1010
1012
Inte
nsity
(cp
s)
Scattering angle (°2)
30 35 40 45 50
102
103
104
Inte
nsi
ty (
arb
.un
its)
Diffraction angle (o2)
Virgin 4h/300°C XRR XRD XRR XRDt(Fe) 69.7 63.5 69.9 61.8t(Au) 20.4 24.3 19.4 25.8t(int) 2.2 2.1 90.1 90.0 89.3 89.7d(Fe) 2.036 2.027d(Au) 2.339 2.327 0.076 0.040(Fe) 8.0 4.5 12.0 6.5(Au) 9.5 5.0 13.0 7.5(surf) 12 20(Fe1) 1.0 0.6c(%) 90 100 85 80
Fe/Au (70Å/21Å)13
29 Physikalisches Kolloquium TUC, 18.6. 2003
Fe/Au (70Å/21Å)13Low correlation of the interface roughness
-6000 -4000 -2000 0 2000 4000 60002000
4000
6000
Sample inclination from the normal direction (arcsec)
Diff
ract
ion a
ngle
(a
rcse
c)
Weak maxima of the resonant diffuse scattering
Small difference between (XRR) und (XRD)
30 Physikalisches Kolloquium TUC, 18.6. 2003
Comparison of the Scattering Phenomena
Continuous Interfaces
XRR Total External Reflection Kiessig Oscillations Bragg Peaks
SAXS Yoneda Peaks Resonant Diffuse Scattering
WAXS Satellite Peaks
Non-continuous Interfaces
XRR Total External Reflection Kiessig Oscillations Bragg Peaks are weaker
SAXS Yoneda Peaks are weaker Resonant Diffuse Scattering is
concentrated at qx=0
WAXS Satellite Peaks are overlapped by the
Diffraction Peak from Matrix
31 Physikalisches Kolloquium TUC, 18.6. 2003
Acknowledgement
Deposition of Fe/Au multilayers Prof. R. Krishnan and Prof. A. Das, CNRS Meudon/
Paris (F)
Transmission electron microscopy Prof. J. Zweck, University of Regensburg (D)