XRD
polykrystalické tenké vrstvy
• Conventional Bragg-Brentano symmetric geometry – θ/2θ scan
• Asymmetric BB geometry – θ/2θ scan
• Parallel beam geometry – 2θ scan
Phase analysis
Lattice parameters
Size, strain
Texture
Bragg-Brentano conventional powder diffraction geometry
Symmetric  - 2 scan
3
h3k3l3
2 h2k2l2
1 h1k1l1
Information from the grains
oriented with the
corresponding planes
parallel to the surface
Absorpce

a
b
b
Lineární absorpční koeficient
Energie z hloubky t za 1 s
dI  I 0 A
2



ab
sin 
2
sin 
2 sin  cos 
sin 2   sin 2 
2
sin  cos
cos 2  1
cos 2
Ae
 Gt
sin   sin b
G
sin  sin b
Asymmetric powder diffraction geometry
3 h k l
3 3 3
h2k2l2
2
2 scan
1 h1k1l1
Small constant angle
of incidence
g
Parallel beam
g = 2 – 10 
Picture from Seifert poster
XRD Seifert - FPM
Monochromator
Detector
Parallel plate
collimator
Slits
X-ray tube
Sample holder
C. Bragg-Brentano asymmetric powder diffraction geometry
3 h3k3l3
hkl
2 2 2 2
1 h1k1l1
 - 2 scan
goniometer
Ygoniometer
Texture
Stress
Philips X’Pert MRD
Eulerian cradle
Sample
stage
X-ray tube
Parallel plate
collimator
Detector
Goebel mirror
Polycapillary
Texture and Stress
Omega sken
PbTiO3
 s can
500
I(cps)
400
300
FWHM
200
100
0
5
Korekce na absorpci a defokusaci
10
15
20

25
30
 - sken
Texture, stress
Pole figures
515 nm
935 nm
2000 nm
Pole figure (100) for the samples of different thickness. The asymmetry of the texture
(left, middle) as well as the inclination of the texture (right) can be seen in 2.5D plot.
2D reciprocal space scan
2scan
2
2 scan
 scan
Ideal single crystal
Ideal polycrystal
Textured polycrystal
0

Zbytková napětí
Homogenní napětí 1. druhu (s).
d 1 hkl
 s2 s sin 2   2 s1hkls
d
2
Může být určováno přímo známou metodou sin2,
kdy musí být vzorek nakláněn na různé úhly  ze
symetrické polohy tak, aby difraktovaly atomové
roviny různě skloněné vůči povrchu. Uvedený výraz
platí přesně pouze pro jednoosá napětí (0 pro
symetrickoul Braggovu-Brentanovu geometrii).
Rtg elastické konstanty
n
1 n
s1   , s2 
E
E
n… Poissonovo číslo, E … Youngův modul
Elasticky izotropní materiály
tlakové napětí
Elastická anizotropie +
Reussův model ( skonst. 
maximální závislost na hkl )
a
s1R ,hkl  s12  s0 ,
1 R ,hkl
s2  s11  s12  3s0
2
h 2 k 2  h 2l 2  k 2l 2

, s0  s11  s12  0.5s44
(h 2  k 2  l 2 ) 2
222
400
311
311
111
200
a0
Hodnota bez napětí
cos cot 
Back
2  sken
422
422
 goniometr
 goniometr
Crystallite Group Method
BB - 0
BB - 0
For thin films and some bulk materials the orientation of grains with respect to the
surface may be very important. Differently oriented grains can have very different defect
content and/or be in very different stress state.
Therefore it is desirable to measure various crystallite families (texture components)
rather than individual planes. Of course, as it is not the case of single crystals, other
crystallites always contribute to the profile (less for strong texture).
Hloubka průniku
t 
Nekonečná tloušťka
I
 dI dt
t
t 0
Hloubka průniku
Efektivní hloubka
průniku
1 1
t  ln 
r 
Poměr energií difraktovaných tenkou
vrstvou
na povrchu a tenkou vrstvou v hloubce t
1 1
ln ; dI ( z   ) / dI ( z  0)  r
G r
r 
e
r e
Informační hloubka
1
te  Gt
i   z dz /  Adz 

G 1  e  Gt
0
0
Přispívající tloušťka
I (z   R )
R
I (z  t)
Ekvivalentní tloušťka
1 e Gt
 eq 

G G
t
t
R 
1 
1
ln
G  (1  R)  Re Gt



 - 2
(BB)
Hloubka průniku
2
(SB, PB)
Rutile
P42/mnm
4.5977
4.5977
2.9564
b: 5.44900
a: 3.77100
Anatase
I41/amd
3.7710
3.7710
9.430
Brookite
Pbca
9.174
5.449
5.138
4 00
P o wde r S im u la ti o n
3 50
Rutile
3 00
2 50
2 00
1 50
1 00
50
0
2 0. 0
2 5. 0
3 0. 0
3 5. 0
2 
4 0. 0
4 5. 0
5 0. 0
3 50
P o wde r S im u la ti o n
Anatase
3 00
2 50
2 00
1 50
1 00
50
0
2 0. 0
2 5. 0
3 0. 0
3 5. 0
2 
4 0. 0
4 5. 0
5 0. 0
1 80
P o wde r S im u la ti o n
Brookite
1 60
1 40
1 20
1 00
80
60
40
20
0
2 0. 0
2 5. 0
3 0. 0
3 5. 0
2 
4 0. 0
4 5. 0
5 0. 0
Parallel beam geometry
Bragg-Brentano symmetric geometry
Thickness - 0.6 m
3 deg, 100 C
3 deg, 300 C
BB, 300 C
I(cps)
600
Anatase
400
Amorphous
200
0
10
30
50
2
70
Williamson-Hall plot
0.008
Williamson-Hall plot
Integral breadth (1/d)
0.007
0.006
0.005
0.004
h = 800 nm
ann. 300 oC
Crystallite size
> 100 nm
Microstrain
~ 0.15 %
0.003
~ microstrain
0.002
0.001
~ 1/crystallite size
sin 
0
0
0. 1
0.2
0.3
0.4
0.5
0. 6
0.7
0. 8
0.9
1
measurements from both XRD7 and X’Pert diffractometers
2and-2 scans
b hkl (1 / d ) 
1 4ehkl

sin 
Dhkl

Apparent crystallite size
Lattice strain e=d/d
Texture indices
Thicker
Thinner
2.0/250 2.0/300 1.7/300 1.5/300 1.2/300
101
1.2
1
1.2
1.6
1.7
004
3
2.1
1.3
1.1
1.1
112
0.5
0.5
0.8
1Fiber texture 0.9
200
1.2
0.9
0.7
1
1.1
105
1.6
1.2
1.1.
0.9
1
211
0.6
0.5
0.6
0.9
0.9
Residual stress
Stress: 383,5 ± 19,9 MPa
Phi = 0,0°
1.2654
1.2652
1.2650
•
1.2648
d-spacing (A)
1.2646
1.2644
1.2642
1.2640
•
•
•
•
isotropic elastic constants
(E = 190 GPa, ν = 0,31)
tensile stress
at 500 C drop of stress
stresses ~ 200 - 300 MPa
typical stress anisotropy
1.2638
1,54 m at 300 C for (215)
1.2636
1.2634
1.2632
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
sin ² (Psi)
Typical linear dependence
Isotropic stress, absence of tri-axial stresses
Stress
d(105) [A]
1.701
Thick
ness
[nm]
1.5m, 300°C
1.5m, 350°C
1.700
s
[MPa]
300
C
200
1.699
350 C
500 C
341
151
630
187
187
42
800
219
209
-
1000
184
154
-
1500
240
163
-
1700
280
232
-
2000
293
252
-
1.698
0.0
0.2
0.4
0.6
sin 
2
0.8
Stress anisotropy
0.2 m, 300°C
0.8 m, 300°C
0.8 m, 350°C
1.5 m, 300°C
1.5 m, 350°C
0.005
0.004
0.003
0.002
0.001
diffraction peak
(215)
(220)
(116)
(211)
(105)
(200)
(112)
(004)
0.000
(103)
2
slope of sin  plot
0.006
105
Diffraction peaks
For different  inclinations
211
300 ºC
Tensile stress
~ 200 MPa
500 ºC
no stress
X-ray reflectivity
Refraction index
n 1
Total reflection
n  1  (  ib ),  , b ~ 105
2

re  e
2

b

4
electron density
absorption length
re = 2.818  10-15 m
 - wavelength
n  1
r0
N at ( f1  if 2 )
2
2
c  2
Critical angle
Surface roughness, film thickness
R  R0 exp(16 2s 2 g sin i / 2 )
10
0
7
10
~ 1/t
t=0,054m
6
10
10
-2
Perfectly smooth surface
Reflectivity
Reflectivity
10
-1
5
10
Visible up to ~ 300 nm
4
10
3
10
10
10
-3
2
10
1
10
-4
Kiessig maxima
2t sin 2  im  sin 2  c  m
0
10
10
-5
1
0.3 nm roughness
1
2
angle (deg)
3
2 (degrees)
Reflectivity is sensitive only
to the projection of the surface profile
to its normal direction
2
It cannot distinguish between
mechanically and chemically rough
surface
3
TiO2 200 nm
250 ºC
350 ºC
450 ºC
Increasing roughness with annealing temperature
TiO2 200 nm
250 ºC
Ω scans
TiO2 1 700 nm
350 ºC
Ω scan
Reflectivity curves
Reflectivity
t= 1.0 m
t= 1.7 m
t= 0.054 m
t= 1.24 m
6
10
5
10
4
10
Reflektivita
as deposited
o
350 C
o
450 C
6
10
7
10
5
10
4
10
3
10
3
10
2
10
2
10
1
1
10
0
10
10
0
10
-1
10
-1
10
-2
10
-2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
úhel 2
(stupně)
Increase of roughness with film thickness
3.5
10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
2
Reduction of very thin surface layer with
annealing temperature
4
Reflectivity curves fitting
Two layer model necessary
Surface porous layer
100,000
0,8 m 300 C
10,000
1,000
100
0,054 m 350 C
100,000
10
10,000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Incident angle (°)
Intensity (counts/s)
1
0.7
1,000
0.8
0.9
1
1.1
100
10
Experimental
1
Fitted
0
0
0.1
0.2
0.3
Incident angle (°)
0.4
0.5
Surface roughness
Thickness
[nm]
Fitted
thickness
[nm]
Electron
density
[g.cm-3]
Roughness [nm]
54
57
3.42
1.2
100
93
3.58
1.7
200
200
3.48
1.8
630
569
3.53
4.1
57
3.64
7
968
3.89
2
49
3.68
5
1791
3.89
2
55
3.74
5
2nd layer
1000
2nd layer
1700
2nd layer
Depth profiling
thickness – 1 m
Different angles of incidence ()
Rutile 110
0.5
0.75
1.0
1.5
2.0
Anatase 101
Rutile growths on the interface while anatase is on the top
thickness 935 nm
Anatase
Angles of incidence in degrees
Effective penetration depth
0.5
0.75
1.0
1.5
2.0
100 nm
150 nm
200 nm
300 nm
400 nm
Rutile
Rutile growths on the interface while anatase is on the top
Reflection on multilayers
Bragg maxima of multilayer
Period d
k  2d n 2  cos 2  i
d  d A  dB
d
k
2(sin 2  sin 1 )
d=T
Kiessig maxima
Total thickness T
Number of
ML periods
Annealing of amorphous
9x(5 nm Si/ 1 nm W)
4
10
3
10
2
10
1
10
Reflectivity
10x(GaAs 7nm/AlAs 15 nm),
CuK1
Kinematical approx:
No total reflection region,
wrong positions of the satellites
(refraction not considered)
0
10
10
-1
10
-2
10
-3
1x10
-4
1x10
-5
10
-6
1
2
3
2 (degrees)
4
Experimental set-up
Detector
X-ray tube
CuK
Göbel mirror
Slit 0.05 mm
Secondary
graphite
monochromator
Slit 0.1mm
Sample
Diffuse scattering
non-specular conditions
Thermal fluctuations
Correlated layer distortions
Height-height correlation
function
C( X , Y )  z( X , Y ) z(0,0)
C( R)  s 2 exp(R /  )2h
Effective cut-off length of the
self-affine surface
For multilayers
C jk ( X , Y )  z j ( X , Y ) z k (0,0)
C jk  C j Ck exp(  | Z j  Z k | /  )
Vertical interface roughness
correlation
Fe/Au (70Å/21Å)x13
Low correlation of the interface roughness
-1.11
-0.56
0
-0.56
1.11
1.67
6000
3.33
4000
2.22
1.11
2000
-6000
-4000
-2000
0
2000
Sample inclination
4000
6000
Detector angle
-1.67
Dynamická difrakce
Dynamical diffraction
Shift from the kinematical Bragg
position (due to refraction)
Finite width of the diffraction curve
(even for T→0)
Asymmetry of the maximum – due
to the Borrmann effect
Wavefields in crystal
Weakly interacts with the atoms –
Anomalously low absorption
Strongly interacts with the atoms –
Anomalously high absorption
The Borrmann effect
Epitaxní vrstvy
strain
Tloušťka
Implantace
Si – B+
D = 3,1.1014
D = 6,2.1015
D = 6,2.1015
a žíhání 1000 ºC
bez implantace
X-ray grazing incidence diffraction
W ~ 1.8 nm na Al2O3
a|| = 0.3184 nm
a0 = 0.3165 nm
e|| = 0.6 %
<D||> 5 nm
 sken
Mozaiková rozorientace ~ 1.1º
MBE
Mo 22 nm (111)
na
(001) GaAs
Tři domény
Mo[110] || GaAs [110]
GaAs [1-10]
GaAs [100]
Mismatch
B || -10.2 %
┴ +3.7 %
C ┴ +27 %
<D> ~ 13 nm
Jedna doména
Nb[110] || GaAs [100]
Nb(001) || GaAs (001)
Mismatch
21.1 %
Standing waves
Standing waves
 2


 2  | Eh |


| Eh |
I (r )  | E0 | 1   2  2  cos(h r   )
| E0 |
 | E0 |

Amplitude of diffracted wave
Phase of (Eh/E0)
Amplitude of incoming wave
Reciprocal lattice vector
Reflection curve – 1
Phase – 2
Intensity at atomic planes – 3
Maximum interaction
for  = 0, at high angle
side of reflection curve
Monolayer of adsorbed atoms
High sensitivity to
displacement of layer
~ 1 % !!!
Yield of the fluorescent radiation
Adsorbed layers
Three adatoms
at 0, 1/3, 2/3
Parallel planes
Inclined planes
Experiment
Measurements of secondary radiation
under the condition of diffraction
Fluorescence
Photoelectrons
Auger electrons
Compton radiation
Chemical selectivity
Spatial resolution on atomic scale
Depth-resolved studies
Determination of coherent position – mean plane of the adsorbed atoms
and coherent fraction – static and dynamic displacement of atoms from the coherent position
Organic materials
Long-period standing waves are necessary
Bragg diffraction from layered synthetic microstructures with large period
10-200 layer pairs (low and high electron density)
Fixed period XSW
Total reflection
SW is formed as an interference between incident
and specularly reflected waves
 = 0.1 c
 = c
Height dependence of electric field intensity
generated during specular reflection of
an X-ray plane wave from the mirror surface
at three angles of incidence
Marker atom A – two E-field maxima
marker atom B – five E-field maxima
XSW application
Monitoring of membrane-related dynamic processes
membrane-lipid phase transitions
ion movement in membrane
Protein folding
Membrane-protein insertion
Lipid and/or protein distributions
Surface binding
Distribution of marker atom above the substrate surface
Theoretical model  Experimental fluorescence,
Reflectivity data
Layered model of refractive index
based on the known structure
Adjusting of interfacial roughness
Features of XSW
Resolution ~ 1 % of LSM d (for 50 Å - 0.3 Å, 925 Å - 3 Å)
Element specificity (not suitable for light elements, O, P, S)
Structure-determination measurement on isolated lipid membranes
(protein monolayers ~ 10 pmol (100 ng) of cytochrome c)
Calculation of the angle-dependent electric-field profile
and fluorescence-yield profile normal to the surface
Adjusting two parameters
Membrane-topology measurements on minimally perturbed systems
(Fe XSW on Fe-cytochrome c)