Röntgenbeugung und
Röntgenstreuung an
Multilagenschichten mit
diskontinuierlichen Grenzflächen
David Rafaja
Institut für Metallkunde
Struktur und Gefüge von Werkstoffen
TU Bergakademie Freiberg
http://www.ww.tu-freiberg.de/mk/
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)
2
Physikalisches Kolloquium TUC, 18.6. 2003
Microstructure Model
Fe/Au-Multilayer
Anticipated changes of the
multilayer microstructure (after a
temperature treatment)
10 nm
TEM courtesy of Prof. J. Zweck, University of Regensburg
3
Physikalisches Kolloquium TUC, 18.6. 2003
Real Structure of Multilayers
From SAXS (small-angle X-ray
scattering)
From WAXS (wide-angle X-ray
scattering)




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)
 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
 Interface continuity
4
Physikalisches Kolloquium TUC, 18.6. 2003
Experimental set-up
Used for XRR,
SAXS, GAXRD and
symmetrical XRD
Angle of
incidence, g
Sample
Sample
inclination, y
Diffraction
angle, 2q
Scintillation
detector
Goebel mirror
Sample
rotation, f
Diffraction vector
Normal direction
Flat monochromator
5
X-ray source
Physikalisches Kolloquium TUC, 18.6. 2003
Can the interface discontinuities
be seen by X-rays?
X-Ray Reflectivity
Theoretical background
- Multiple (dynamical) scattering of X-rays
- Optical theory for smooth interfaces (no
interface roughness)
Recursive formula
E Rj1
E j 1
 A j 1 
r j , j 1 
Substrate
qj 
4

r j , j 1  f j2 A j
1  r j , j 1 f j2 A j
q j 1  q j
q j 1  q j
sin q j 
f j21

f j  exp  iq j t j 2
;
4


n 2j  cos 2 q
re 2 at
n j  1
N j f 0 j  f j  if j  1
2


Based on: L.G. Parrat, Phys. Rev. 95 (1954) 359.
7
Physikalisches Kolloquium TUC, 18.6. 2003
I  RN
2
X-Ray Reflectivity
Theoretical background
The interfaces must be continuous
j-2
X-ray reflectivity of multilayers with
a certain interface roughness
E Rj1
E j 1
tj-1
 A j 1 
r j , j 1 
r j , j 1  f j2 A j
1  r j , j 1 f j2 A j
q j 1  q j
q j 1  q j

tj

f j21
exp  q j 2j , j 1 2

f j  exp  iq j t j 2 ; q j 
4


n 2j  cos 2 q
Change in the Fresnel reflection
coefficient (Debye-Waller factor)
Substrate
L. Névot, P. Croce, Rev. Phys. Appl. 15 (1980) 761.
DWBA
8
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.
Physikalisches Kolloquium TUC, 18.6. 2003
X-ray Diffuse Scattering
on continuous interfaces
Distorted wave Born approximation - DWBA
Differential cross-section of the
diffuse scattering

S k02 1  n 2
 d 

 d 
diff
16 2
t j , j 1 
~
F
Substrate
C (x,y) … In-plane correlation of
interface corrugations
In a multilayer: additionally the
vertical correlation
9

2k j
k j  k j 1
2
2
2~
t j , j 1 kin  t j , j 1 kout  F



exp k j  k j 1 2  2j 8
 q C  x, y   i xq x  yq y 
dxdy e z
 1e
S



2
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.
Physikalisches Kolloquium TUC, 18.6. 2003
Intesity (a.u.)
X-ray Reflectivity
10
8
10
7
10
6
10
5
10
4
10
3
Structure model
, t,  (top)
, t,  (X)
B ra g g re fle c tio n s
, t,  (C)
, t,  (B)
, t,  (A)
,  (S)
K ie s s ig o s c illa tio n s
10
2
10
1
10
0
1
2
3
4
5
o
Glancing angle ( 2 q )
10
6
7
qx  q y  0 ; qz  0
Capping layer
Layer X
Layer C
z
Layer B
Layer A
Substrate
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.
Physikalisches Kolloquium TUC, 18.6. 2003
XRR Curve of a Periodic Multilayer
Au/Al, 10x, t A +t B =7.5nm
10
8
10
6
10
4
10
2
10
0
t(A)/t(B)=1/1
t(A)/t(B)=1/2
Reflectivity
t(A)/t(B)=1/3
t(A)/t(B)=1/4
 Total reflection  Electron density of the
uppermost layer
re2
n  1
e  f 0  f   if   1
2
 Decrease of the reflected Intensity  interface
roughness
I  q 4 exp  q 2  2 2
 Kiessig oscillations  thickness of the whole
multilayer
2
2
qt  4 t

10
-2
10
-4
10
-6
10
-8
n
 cos q  2m
 Bragg-like peaks  thickness of the periodic
motif
2
2 n  cos 2 q  m
 Extinction of the Bragg-like peaks  thickness
of the individual layers in the multilayer system
0
2
4
6
8
10
o
Glancing angle ( 2 q )
11
Physikalisches Kolloquium TUC, 18.6. 2003
I n  t B t A  1  0
X-ray Diffuse Scattering of a
Periodic Multilayer
Reciprocal space mapping
Observed phenomena
Q/2Q (arcsec)
 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
Sample inclination (arcsec)
8000
6000
 V.Holý, T.Baumbach, Phys. Rev. B 49 (1994)
10668.
4000
-6000
-4000
-2000
0
2000
4000
6000
Information on the
mesoscopic Structure in
the lateral direction and
on the vertical correlation
of disturbances
qx-qz scan at qy = 0
Coplanar diffraction
geometry
12
Physikalisches Kolloquium TUC, 18.6. 2003
Fe/Au Multilayers
Intensity (a.u.)
Experimental example
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
0
Fe/Au (27Å/23Å)x10
Si/Au(100Å)
Refined parameters
t (Fe)
(27 ± 2) Å
t (Au)
(23 ± 1) Å

1
2
3
4
5
6
7
50 Å
 (Fe)
5Å
 (Au)
5Å
 (Fe)
(1.4 ± 0.2)
 (Au)
(0.9 ± 0.1)
o
Glancing angle ( 2Q)
13
Physikalisches Kolloquium TUC, 18.6. 2003
Binary System Fe – Au
A ssessed A u - F e p h ase d i ag r am . C al cu l at ed .
Au
14
Fe
Physikalisches Kolloquium TUC, 18.6. 2003
XRR on Multilayers with NonContinuous Interfaces
Interfaces
Continuous
Aj 
Regions
f j21 A j 1  r j , j 1
f j21 A j 1r j , j 1  1

f j2  exp iq j t j
r j , j 1 
qj 
4

Continuous

q j  q j 1
q j  q j 1
exp

 q j q j 1  2j
2

Amplitude and Phase shift
n 2j  cos 2 q
Reflectivity
R  AN
Discontinuous
A j  f j21 A j 1
Aj  c j
15
Discontinuous
2
f j21 A j 1  r j , j 1
f j21
A j 1 r j , j 1  1


f j2  c j f (12) j  1  c j f (22) j
Physikalisches Kolloquium TUC, 18.6. 2003


 1  c j f j21 A j 1
XRR on Multilayers with NonContinuous Interfaces
Reflectivity (arb.units)
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
10
-11
Fe/Au (30Å/10Å) x 8
Simulation
Changes in the XRR curve
 Intensity of Bragg peaks
decreases
 The fringes near the TER are
shifted
(a)
c = 100%
c = 60%
(b)
c = 30%
(c)
0
2
4
6
8
10
Consequences
 The structure refinement using
the classical model yields closer
electron densities of the
alternating materials and larger
roughness of all interfaces
Glancing angle (°2q)
16
Physikalisches Kolloquium TUC, 18.6. 2003
Diffuse Scattering from Multilayers
with Non-continuous Interfaces
DWBA:
Differential cross-section
 d 

 d 
  diff
Continuous
t j , j 1 
~
F
2k j
k j  k j 1



 q C  x, y   i xq x  yq y 
dxdy e z
 1e
S



k x, y   1

16 2
2
2~
t k1  t k2  F
Interfaces
exp k j  k j 1 2  2j 8
2

2
2 2
S k0 1  n
Discontinuous

~
F
 q C  x, y   i xqx  yq y 
dxdy e z
 1e
 k  x, y 
S


k

Form-factor
The integration is performed
only in the continuous regions
17

t j , j 1  c j t j , j 1  1  c j  1
Physikalisches Kolloquium TUC, 18.6. 2003
2
 k x, y   0
Intensity (cps)
Diffuse Scattering from Multilayers
with Non-continuous Interfaces
10
7
Consequences
10
6
10
5
Decrease of the intensity of the
Yoneda peaks  modified
Fresnel transmission coefficients
10
4
10
3
10
2
10
1
10
0
Broadening of the specular peak
in the longitudinal scans 
„convolution“ with the formfactor
D. Rafaja, H. Fuess, D. Šimek, J. Kub, J. Zweck,
J. Vacínová, V. Valvoda, J. Phys.: Condensed
Matter 14 (2002) 5303-5314.
-1.0
-0.5
0.0
0.5
1.0
Sample inclination (deg)
18
Physikalisches Kolloquium TUC, 18.6. 2003
Diffuse Scattering from Multilayers
with Non-continuous Interfaces
Fe/Au (70Å/21Å)13 / 280Å Au / SiO2
12
10
10
10
10
10
8
10
6
10
4
10
2
10
0
Intensity (cps)
Intensity (cps)
10
As deposited
2h/200°C
2h/300°C
4h/300°C
0
1
2
3
4
5
6
7
10
8
10
6
10
4
10
2
10
0
-0.9 -0.6 -0.3
0.3
0.6
Sample inclination,  (deg)
Diffraction angle (°2q)
19
0.0
Physikalisches Kolloquium TUC, 18.6. 2003
0.9
Wide-Angle X-ray Scattering
10
Structure model
3
tB
2
Intesity (cps)
10
tA
Intralayer
disorder
Continuous and
discrete interface
roughness
Average d-spacing
Interlayer distance
10
1
 Jahn-Teller-Method (layered structures)
 Additional information on the atomic ordering
(interplanar distances, intralayer disorder, texture)
30
35
40
45
50
o
Diffraction angle ( 2 q )
20
55
E.E. Fullerton, I.K. Schuller, H. Vanderstraeten and Y.
Bruynseraede, Phys. Rev. B 45 (1992) 9292.
Physikalisches Kolloquium TUC, 18.6. 2003
Kinematical Theory of WAXS for
Multilayers with Continuous Interfaces
I  FSL FSL* ; FSL   FL eiqt L  aL 
Intensity:
L
Positions of interfaces
(Gauss-like distribution):
2
1  aL  a  
PaL  


2c 
2c 2 
Positions of individual atoms
(correlated displacements):

r  x, y, nd L  n d :


NL
Structure factor of individual
layers:
Interatomic distances and
their fluctuations:
21
FL   f n e
n 1

iq  r
NL

 f n eiqndL  P0 eiq
n 1
n d

e
 d 2
2
d d  

 
q 2 2 
e N L  1
  f n 
FL   f n exp n iqd L 
4
e 1
n 1

 
q 2 2
  iqd L 
4
NL
Physikalisches Kolloquium TUC, 18.6. 2003
WAXS Diffraction Pattern of a
Periodic Multilayer
Fe/Au (3.24nm/1.41nm)  12
Fe: 16  0.20268 nm, Au: 6  0.2355 nm
2 sin q n
35

-2
-3
30
-1
1
n

d 
Periodicity of a bi-layer:
-4
20

 d
2q0
25
Intensity (a.u.)
Positions of Satellites:
  N Ad A  N B d B
+1
15
Mean interplanar spacing:
10
+2
5
dB
dA
0
30
32
34
36
38
d 
40
42
44
46
48
N Ad A  N B d B


N A  NB
N A  NB
50
o
Diffraction angle ( 2 q )
22
Physikalisches Kolloquium TUC, 18.6. 2003
WAXS on Multilayers with NonContinuous Interfaces
Kinematical Theory
Structure model
Matrix + Precipitates
  
  
  
E   E0 r eiqr dr    M E0 r eiqr dr    P E0 r eiqr dr
V
P
M







 P E0 r eiqr dr     P E0 R j  r  e
P
j
P
  
iq  R j  r 

dr 

 iqR j
iqr  
  E0 R j e

e
 P dr 
 
P
j
buffer
M

 iqR j
 iqr 
 iqr 
iqr  
 M E0 r e dr    M E0 r e dr   E0 R j e

e
 M dr 
E
V
substrate
23
 e
 
V
P
j

 iqR j
 iqr 
iqr  


 M E0 r e dr   E0 R j e



e
dr 
 P M

iqr 
 

dr     e
j
  
iq Pk  r 

dr 
k
Physikalisches Kolloquium TUC, 18.6. 2003
e
k
P
 
iqPk
 e

iqr 
 

iqPk
dr    f k e
k
WAXS on Multilayers with NonContinuous Interfaces
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
 

 
 iqR j
iq Pj , k  

iqPM
 f P  f M e
  e
E  E0   f M e


 
Sample
k
j 


F

iqt
iqz
E  E0 1  c FM  c  e j FAj  FBj e Aj

j
e

FML
iqz j
Aj
 FBj e
iqt Aj



j
E  E0 1  c FM  cFML  ; I  E  E 
I  E0
2
1  c
Matrix
2
FM
2
 c 2 FML
2
Multilayer


 2c1  c  Re FM FML
Interference Term
D. Rafaja, H. Fuess, D. Simek, L. Zdeborova and V. Valvoda, J. Phys.: Condens. Matter 14 (2002) 10021-10032.
24
Physikalisches Kolloquium TUC, 18.6. 2003

WAXS – Simulation of Interface
Discontinuity
20 % Interface discontinuity 40 %
1000
1000
-2
(b)
-1
800
Inte nsity (a .u.)
(a)
800
I  E0 
2

600
600
 1  c  FM
400
400
 c 2 FML
200

 2c1  c  Re FM FML
200
-3
0
1
2
-4
0
32
0
36
40
44
48
32
36
40
44
48
o
Diffra ction an gle ( 2 Q ) Diffra ction an gle ( 2 Q )
o
25
Physikalisches Kolloquium TUC, 18.6. 2003
2
2
2




Combined Refinement
SAXS/WAXS
Fe/Au (26Å/24Å)10
4
Intensity (cps)
10
10
10
8
10
6
10
4
10
2
10
0
Virgin
2h/200°C
XRR XRD
XRR XRD
t(Fe) 26.5 25.6
26.5 27.0
t(Au) 24.0 24.6 22.0 27.8

50.5 50.2 48.5 54.8
d(Fe)
2.031
2.027
d(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
9.0
c(%) 90 100
85
80
0
2
4
6
3
10
2
10
30
35
40
45
o
Diffraction angle ( 2q)
Scattering angle (°2q)
26
Intensity (arb.units)
10
Physikalisches Kolloquium TUC, 18.6. 2003
50
Fe/Au (26Å/24Å)10
Large correlation of the interface roughness
Well-pronounced maxima of the resonant diffuse scattering
Large difference between (XRR) and (XRD)
Diffraction angle (arcsec)
8000
6000
4000
2000
-6000
-4000
-2000
0
2000
Sample inclination from the
normal direction (arcsec)
27
Physikalisches Kolloquium TUC, 18.6. 2003
4000
6000
Combined Refinement
SAXS/WAXS
Fe/Au (70Å/21Å)13
12
10
4
10
10
8
Intensity (cps)
10
6
10
4
10
2
10
0
10
0
2
4
6
Virgin
4h/300°C
XRR
XRD
XRR
XRD
t(Fe) 69.7 63.5
69.9 61.8
t(Au) 20.4 24.3
19.4
25.8
t(int)
2.2
2.1

90.1 90.0
89.3
89.7
d(Fe)
2.036
2.027
d(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
0.6
c(%) 90 100
85
80
Scattering angle (°2q)
28
Physikalisches Kolloquium TUC, 18.6. 2003
Intensity (arb.units)
10
3
10
2
10
30
35
40
45
o
Diffraction angle ( 2q)
50
Fe/Au (70Å/21Å)13
Low correlation of the interface roughness
Weak maxima of the resonant diffuse scattering
Diffraction angle (arcsec)
Small difference between (XRR) und (XRD)
6000
4000
2000
-6000
-4000
-2000
0
2000
Sample inclination from the
normal direction (arcsec)
29
Physikalisches Kolloquium TUC, 18.6. 2003
4000
6000
Comparison of the Scattering
Phenomena
Continuous Interfaces



XRR
Total External Reflection
Kiessig Oscillations
Bragg Peaks

SAXS
Yoneda Peaks
Resonant Diffuse Scattering

WAXS
Satellite Peaks

30
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
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)
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Physikalisches Kolloquium TUC, 18.6. 2003