X-ray analysis methods
Mauro Sardela, Ph.D.
Frederick Seitz Materials Research Lab.
University of Illinois at Urbana-Champaign
X-ray interactions with matter
Incident x-ray photons
hn 0
Coherent
scattering
hn 0
Incoherent
scattering
Fluorescence
hn1
hn 2
e- Photoelectron
e-
Auger electron
2 nm ~30 mm
Interaction
volume
10 mm – 5 cm
hn : photon
e-: electron
Sample
X-ray radiation mostly used in lab instruments:
Cu radiation
g
X
rays rays
UV
Visible
• Cu Ka: l= 0.15418 nm
(8.05 keV, conventional resolution)
• Cu Ka1: (l= 0.15056 nm (high resolution)
2
X-ray interactions with matter
Coherent scattering
(3)
(Diffraction, Thompson or
Rayleigh scattering)
E0
(1)
Incoherent scattering
E0
(Compton scattering)
E0
(2)
Electron
Photon
(1)
(3)
(2)
Nucleus
Incoming photon
Oscillating electron
Scattered photon
No loss of energy.
Fluorescence
E0
(1)
(2)
(3)
E1=EL-EK
(1)
(3)
(3)
(1)
(4)
(4)
(2)
(2)
(1)
(2)
(3)
(4)
(5)
Incoming photon
Energy is partially transferred to electron
Scattered photon (energy loss).
Auger electron
E0
Photon
Nucleus
K
Electron
L
shell
Incoming photon
Expelled electron (photoelectron)
Hole is created in the shell
Outer shell electron moves to the inner shell hole
Energy excess emitted as characteristic photon.
Electron
Photon
Nucleus
(1)
(2)
(3)
E1<E0
(5)
Nucleus
L
shell
K
(6)
Electron
(1-4) Hole created (3) after photoelectron emission (2)
is occupied by outer electron (4).
(5) Excitation energy is transferred to electron
(6) Electron ejected from atom (Auger electron)
Fundaments of diffraction
F
“Real” space
Reciprocal space
Point
Set of planes
(h k l)
d
hkl
2p/d
origin
M. von Laue 1879-1960
X-rays from crystals, 1912.
4
Fundaments of diffraction
F
“Real” space
Reciprocal space
d
2p/d
origin
5
Fundaments of diffraction
“Real” space
F
Reciprocal space
origin
6
Bragg’s law and Ewald’s sphere
F
Reciprocal space
“Real” space
Bragg’s law
Ewald’s sphere
w
2q
q
k1
2q
k0 (= radius)
2 d sin q = n l
1862–1942
1890-1971
Elastic (Thompson’s)
scattering
q = k1 – k0
q: scattering vector
q = (4 p/l) sinq
Paul7 P. Ewald
1888-1985
Diffraction in reciprocal space
Single crystal
X-ray
Poly crystal
(random)
Point detector scan
Diffraction plane
Poly crystal
(texture)
Coupled 2theta-omega scans
q
k1
w
2q
X-ray
source
k0
d
2q
w
Detector
Sample
2q-w scan:
Probes d-spacing variation
Along q
Phases id, composition, lattice constants
Grain sizes, texture, strain/stress
q
k1
2q
k0 (= radius)
Rocking curve omega scans
q
k1
X-ray
source
Detector
k0
w
2q
d
w
Sample
w scan:
Probes in-plane variations
Normal to q
Mosaicity, texture and texture strength
q
k1
2q
k0 (= radius)
Information contents in the XRD pattern
Detector
X-ray
source
2q
w
Sample
Peak position:
identification, structure, lattice parameter
Peak width:
crystallite size, strain, defects
Peak area or height ratio:
preferred orientation
Peak tails:
Diffuse scattering, point defects
Background:
amorphous contents
Powder diffraction methods
Crystalline? Amorphous?
What elements, compounds, phases
are present?
Structure? Lattice constants?
Strain?
Grain sizes? Grain orientations?
Is there a mixture? What % ?
Powders, bulk materials, thin films,
nanoparticles, soft materials.
12
Bragg-Brentano focusing configuration
Receiving
slit
Focus
X-ray
source
Single crystal
monochromator
(l)
Detector
Divergence
slit
Secondary
Optics:
Scatter and soller slits
Focusing
circle
(variable during
measurement)
Divergent beam
not good for
grazing
incidence
analysis
specimen
Angle of
incidence
(w)
Detector
rotation
(2q)
Sample height
positioning is critical
Diffractometer circle
13
(fixed during measurement)
Ground to fine
powder
(random grain
orientations)
+
Added
amorphous
phases (glass)
to complicate
things… 
XRD powder analysis walkthrough
Intensity (sqrt counts)
Crystalline
phases
Amorphous
(zero background holder)
20
30
40
50
60
70
80
2-theta (degrees), Cu K-alpha radiation
90
100
110
Intensity (sqrt counts)
XRD powder pattern
~ 20 w%
amorphous
added
No amorphous added
20
30
40
50
60
70
80
2-theta (degrees), Cu K-alpha radiation
90
100
110
Peak fit and shape analysis
Data + Peak
+
shape
function
Intensity (sqrt counts)
Peak fit:
Instrument
resolution
FWHM = f (2q)
Crystallinity =
ΣPeak areas
= 81.7 %
Total area
Amorphous
= 1 – (crystallinity) = 18.3 %
contents
20
30
40
50
60
70
80
2-theta (degrees), Cu K-alpha radiation
90
100
110
Search / match
Peak position
+ intensity ratio
√
Match! Fingerprinting
identification of phases
ICDD PDF4 database
ICSD, etc.
Search against…
Hits
Formula
FOM
PDF
RIR
Space group
Calcite
CaCO3
1.1
04-012-0489
3.45
R-3c(167)
Dolomite
Ca1.07Mg0.93(CO3)2
15.0
04-011-9830
2.51
R-3(148)
Search / match
Second
round
√
Focus on
unmatched peaks
Hits
Formula
FOM
PDF
RIR
Space group
Calcite
CaCO3
1.1
04-012-0489
3.45
R-3c(167)
Dolomite
Ca1.07Mg0.93(CO3)2
15.0
04-011-9830
2.51
R-3(148)
Search / match
Second
round
Focus on
unmatched peaks
Search / Match
Identify additional
phases (~ > 1 w%)
Hits
Formula
FOM
PDF
RIR
Space group
√
Calcite
CaCO3
1.1
04-012-0489
3.45
R-3c(167)
√
Dolomite
Ca1.07Mg0.93(CO3)2
15.0
04-011-9830
2.51
R-3(148)
Quant: RIR reference intensity ratio
(
Ratio of
)~(
crystalline
phases
Ratio of peak areas corrected
by RIR of each phase
)
RIR ~ I / I corundum
1
Ratio of crystalline phases:
Calcite:
79.2 w%
Dolomite: 20.8 w%
(no amorphous included)
Intensity (sqrt counts)
2
1
1
1 1
1
1
1
30
2
2
40
35
2-theta (degrees), Cu K-alpha radiation
2
45
2
1
50
Rietveld refinement
Non-linear least square minimization
H. Rietveld
(1932-)
For each data point i:
Minimize this function:
Sum over n data points
n data points
p phases
m Bragg reflections for each data i
wi, bi, Kl, Yl,j weight, background, scale factor and peak shape function
Data +
preliminary structure:
Refinement parameters:
Background
Sample displacement, transparency and zero-shift correction
Peak shape function
Unit cell dimensions
Minimize and converge
Preferred orientation
figures of merit/quality:
Scale factors
R
Atom positions in the structure
Atomic displacement parameters
Rietveld refinement
Intensity (sqrt counts)
Peak fit:
Instrument
Data + Peak
+
shape
resolution 80.7 w%
Calcite:
function Dolomite:
FWHM = f (2q)
22.2 w%
Amorphous: 17.1 w%
Peak areas
Crystallite
= 81.7 %
Crystallinity
= Σ size:
Calcite:Total area
56.8 nm
Dolomite:
35.6 nm
Amorphous
= 1 – (crystallinity) = 18.3 %
contents
20
30
40
50
60
70
80
2-theta (degrees), Cu K-alpha radiation
90
100
110
Rietveld refinement
Data + Peak
+
shape
function
Intensity (sqrt counts)
Peak fit:
Instrument
resolution
FWHM = f (2q)
Crystallinity =
ΣPeak areas
= 81.7 %
Total area
_
Amorphous
=
1
–
(crystallinity)
Calcite,contents
CaCO3, hexagonal, R3c (167)= 18.3 %
0.499 nm/ 0.499 nm / 1.705 nm <90.0/90.0/120.0>
20
30
40
50
60
70
80
2-theta (degrees), Cu K-alpha radiation
90
100
110
Rietveld refinement
Intensity (sqrt counts)
Peak fit:
Data + Peak
+
shape
function
Instrument
resolution
FWHM = f (2q)
Crystallinity =
ΣPeak areas
= 81.7 %
Total area
Amorphous
_
= 1 – (crystallinity) = 18.3 %
Dolomite,contents
Ca1.07Mg0.93(CO3)2, hexagonal, R3 (148)
0.481 nm/ 0.4819 nm / 1.602 nm <90.0/90.0/120.0>
20
30
40
50
60
70
80
2-theta (degrees), Cu K-alpha radiation
90
100
110
Crystallite size analysis
Scherrer’s equation:
Size =
k*l
cos (q) * (FWHM)
k : shape factor (0.8-1.2)
l: x-ray wavelength
FWHM: full width at
half maximum (in radians)
Simplistic approximation!
Not accounting for peak
broadening from strain and defects
Directional measurement!
Measured along the
specific direction normal to
the (hkl) lattice plane
given by the 2q peak position
Peak width
(FWHM or integral breadth)
___ Measurement
___ Fit
Peak position 2q
26
6.5o
2 nm Fe3O4 nanoparticle
Intensity(Counts)
Intensity(Counts)
Crystallite size analysis
20 nm
(111) grains
in Cu foil
0.5o
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
Two-Theta (deg)
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
Two-Theta (deg)
Intensity(Counts)
145 nm
Si powder
0.17o
113.5
114.0
114.5
Two-Theta (deg)
115.0
115.5
27
Peak shape analysis
Peak fit functions:
Gaussian
Lorentzian
Pearson-VII (sharp peaks)
Pseudo-Voigt (round peaks)
Measurement
Information from fit:
Fit
•Position
•Width (FWHM)
•Area
•Deconvolution
•Skewness
___ Measurement
___ Fit
28
Correction for instrument resolution
Use FWHM curve as a function of 2q
from standard sample (NIST LaB6):
specific for each diffractometer
FWHM: b
b D = (bmeas)D – (binstr)D
D: deconvolution parameter
Measurement
Instrument function
D : 1 (~ Lorentzian)
b = (bmeas) – (binstr)
D : 2 (~ Gaussian)
b2 = (bmeas)2 – (binstr)2
D : 1.5
b1.5 = (bmeas)1.5 – (binstr)1.5
29
Potential artifacts in size determination
For this calculation assume:
• Instrument resolution ~ 0.15o
• 2q = 40o
• Cu radiation
FWHM: b
b D = (b meas)D – (b instr)D
Measured
peak width
(o)
Size, nm
D=1
(Lorentzian)
Size, nm
D = 1.5
Size, nm
D=2
(Gaussian)
0.30
56.4
38.0
32.6
0.50
24.2
19.2
17.7
0.75
14.1
12.0
11.5
1.00
10.1
8.8
8.6
1.50
6.3
5.8
5.7
2.00
4.6
4.3
4.2
Large difference
(up to 48%!) for
narrow peaks (large sizes)
Smaller difference
(~ 10%) for
broad peaks (small sizes)
30
Strain effects in diffraction lines
d
No strain
Macrostrain
uniform tensile or
compressive stress
(lattice expansion or
contraction)
Microstrain
nonuniform strain
(both tensile and
compressive stresses)
(lattice distortion).
Dislocations, vacancies,
defects, thermal effects.
D(2q)
Peak position
shift
(lattice constant
change)
Uniform strain
FWHM
Nonuniform strain
Peak width
change
(symmetric
broadening)
31
Size and strain in peak shape analysis
FWHMstrain = 4 * (strain) * tan q
(FWHM)*cos(q) = kl/(size) + (strain)*sin(q)
Williamson-Hall
Method
Acta Metall. 1 (1953) 22.
(FWHM)*cos(q)
Intercept ~ 1/(size)
Slope ~ micro strain
sin(q)
32
X-ray parallel beam methods
Rough, irregular surfaces
Film / Substrate systems
a
Near surface
region
Glancing / grazing angle applications.
Phase, stress gradients (depth profiles)
33
Parallel beam configuration
Negligible sample
displacement issues
(rough and curved
samples OK)
X-ray
source
Specific optics to maximize
intensities
Primary
Optics:
Parallel plates
collimation
mirror, slits, lens
Single crystal
monochromator
(l)
Detector
rotation
(2q)
Parallel
beam
Angle of
incidence
(w)
Detector
specimen
Excellent for glancing angle
(fixed w) applications
34
Thin film orientation analysis
TaN
Example:
<001>
MgO
TaN film
2
-TaN 002
-TaN
(002)
TaN
MgO(001)
substrate
<110>
MgO
(002)
MgO 002
Intensity (a.u.)
-TaNx/MgO(001)
MgO
Cube on cube
epitaxy:
(001)TaN//(001)MgO
(100)TaN//(100)MgO
40 eV, w = 0.95°
<110>
30 eV, w = 0.75°
20 eV, w = 0.60°
(220)
f scans
-TaN
1.17
/MgO(001)
t = 500 nm
Ts = 600 °C
Intensity (a.u.)
2q/w
scan
(surface
normal
direction)
t = 500 nm
Ts = 600 °C
Ji/JTa = 11
fN = 0.125
f scan (in-plane
surface direction)
<001>
-TaN
-TaN
8.5 eV, w = 0.68°
41.0
41.5
42.0
42.5
2q (deg)
43.0
43.5
Data:
Shin,
Petrov et al,
UIUC
MgO
MgO
0
50
100
150
200
f (deg)
250
30035 350
Glancing angle x-ray analysis
Detector
X-ray
source
w
2q
2q
(fixed)
surface
normal
w
Sample
grains
w constant (~ 0.2 – 4o)
: conventional Bragg-Brentano configuration
2q-w scans probe only grains aligned
parallel to the surface
+
: parallel-beam glazing incidence configuration
2q scans probe grains in all directions
X-ray penetration depth vs. angle of incidence
Low angle region
-Type of radiation
- Angle of incidence
- Material (Z, A, r, m)
37
Regular 2q-w scan vs. glancing angle 2q scan
Regular 2q-w scan
Glancing angle 2q scan
w (fixed,
small)
Probe depth:
Variable (deep)
Constant (shallow)
Regular 2q-w scan
Glancing angle 2q scan
Grain orientations
Directions  to surface
Various directions
Depth resolution
Constant, many mm
• From few nm to mm
Depth profiling possible by
varying angle of incidence
• Sensitive to surface
• Ideal for ultra-thin layers
Best configuration
Bragg Brentano
Parallel beam
Parallel beam (less sensitive to
sample displacement)
38
Glancing angle x-ray analysis
Example: Poly-Si gate in CMOS
Poly-Si
(~ 100 nm)
Substrate
Substrate
Si(001)
substrate
Glancing
angle
Conventional
2q/q XRD
2-theta (degrees)
39
Texture and preferred orientation methods
Anisotropy in grain orientation distribution
What is the preferred orientation?
% of random grains?
Strength / sharpness of the texture?
Crystallographic relationship
between film and substrate?
40
Determination of preferred orientation
Method
Measurement
Principle
Results
Lotgering
factor
(Lhkl)
2q-w scans
Compare Ipeak or Apeak with
expected values from random
samples (PDF)
Lhkl as measure of
texture strength
MarchDollase
(MD)
2q-w scans
Use Ipeak or Apeak with MD
formalism
% of grains that are
more oriented along a
specific direction
Rocking
curve
w scans
Measure FWHM from w scan
for a particular (hkl)
FWHM decreases with
stronger texture
Pole figure
f scans at
various tilts y
Pole plots of intensities from a
particular (hkl)
Texture distribution for
a single (hkl)
Orientation
Distribution
Function
(ODF)
Pole figures
from various
(hkl) ’s
Calculate ODF from various
pole figures with background
and defocussing correction
% of grain orientation
distribution in all
directions (Euller
angles).
41
Pole plot projections
Pole
Equatorial
plane
Wulff projection
(stereographic or
equal-angle projection)
Schmidt projection
(equal-area projection)
S
A
Side view
(normal to
equatorial plane)
y
y
r
r
O
AB = AS
B
Side view
(normal to
equatorial plane)
O
W
Equatorial
plane
Top view
(parallel to
equatorial plane)
O
W
f
f
O
OW = r tan(y/2)
S
Top view
(parallel to
equatorial plane)
OS = √2 r tan(y/2)
42
Basics of pole figure analysis
Surface
normal
f
f
hkl
-1 0 0
(100) Pole figure
y
010
001
0 -1 0
Azimuth
f= 0, 90o,
180o, 270o
Tilt
y= 0, 90o
y
Example: (100) cubic crystal
y: [100],[100]
100
(111) Pole figure
f
-1 -1 1
-1 1 1
y
Pole figure plot
f
111
y
Tilt y= 54.7o
1 -1 1
y: [100],[111]
hkl
(110) Pole figure
f
-1 -1 0
y
0 -1 1
y(radial / tilt)
f (azimuthal rotation)
Azimuth
f= 45o, 135o,
225o, 315o
-1 1 0
-1 0 1
011
101
1 -1 0
Azimuth
f= 0, 90o, 180o,
270o,45o, 135o,
225o, 315o
Tilt y= 45o,90o
110
43
y: [100],[110]
X-ray pole figure analysis of textured materials
Texture results from a rolled Cu foil
Pole figures
y
f
• Texture orientation and quantification.
• Volume fraction of textured grains,
twinning and random distributions.
• Texture strength and sharpness.
• Crystallographic orientation.
• Crystallographic relationship between
layers and substrate.
2q
F2
Orientation
distribution
function (ODF)
sample
detector
f
w
y
(111)
F
Inverse
pole
figures
y
f
Data: Sardela, UIUC
F1
F
44
Residual stress analysis methods
Residual stress?
How much? (MPa – GPa)
Type?
Direction (s)?
Stress gradients?
XRD measures strain (Dd)
Hooke’s law
Elastic properties (E,n)
Stress tensor
45
X-ray analysis of residual stress
Film Normal
stressed
a
y
Lf,y
Incident
X-rays
y
Reflected
X-rays
ay
q
sf
a0
film surface
Tilt y:
o
+60
Intensity (a.u.)
unstressed
o
+45
o
+30o
+15
0
o
-15
o
-30o
-45
o
-60
Diffracting planes
Interplanar spacing d (Angstroms)
79
80
81
82
83
84
o
Dd
__ s (1+n)
____ sin2y
d =
E
1.173
1.172
• Sin2y method. Linear for rotationally
symmetric biaxial stress where the only
non-zero components are s11 = s22 = s//
• y and w scan methods.
1.171
1.170
Slope:
1.169
2-theta ( )
• Quantification of residual stress.
• Compressive (-) and tensile (+) stress.
• Crystallographic orientation of stress.
Stress = -199 MPa
(compressive)
1.168
0.0
0.2
0.4
0.6
0.8
sin2(y)
Stress results from a steel sample
Data: Sangid et al, UIUC
• Glancing angle method (texture).
• Determination of stress tensor.
• Requires crystallinity (no amorphous).
46
85
High resolution XRD methods
Single crystals:
g
a
b
b
a
Accurate measurements of a, b, c, a, b , g
Detailed peak shapes: defects, mosaicity.
c
Film / substrate epitaxial systems:
Measure small variations Da, Dc,… (~ 10-5).
c + Dc
Measure layer tilts Df, ...
Df
Detailed peak shapes: defects, strain, mosaicity.
c
a
47
Instrument resolution in reciprocal space
Beam angular divergence,
detector acceptance and
diffractometer sampling volume
w
sample
Angular
acceptance
of the
secondary
optics
2q
Diffractometer
sampling
w
Primary
beam
divergence
(from primary
optics)
volume
Ewald sphere
48
Instrumentation: high resolution configuration
f
y
4-reflection
Ge(220)
monochromator
x-ray
mirror
x-ray
source
slit
w
slit
3-bounce
analyzer
crystal
2q
Triple axis
detector
Dq =12 arc-sec
Dl/l = 5x10-5
Or:
(triple axis:
12 arc-sec acceptance)
Open detector
(open: < 1o acceptance)
Line detector
1D mode for ultra-high speed
49
High resolution x-ray analysis
•Lattice distortions within 10-5.
Single crystals
Epitaxial films
Heterostructures
Superlattices
Quantum dots
• Rocking curve analysis.
• Film thickness.
• Strain relaxation and lattice parameter measurements.
• Alloy composition and superlattice periods.
• Interface smearing in heterostructures (dynamical simulation).
SiGe / Ge superlattice
InAs / GaAs multilayer
Data: Sardela, UIUC
60
62
64
o
2-theta / omega ( )
66
68
Data: Zhang et al
Log intensity (a.u.)
Log intensity (a.u.)
Log intensity (a.u.)
Data: Wu et al, UIUC
58
InAs quantum dots on GaAs
62
64
66
2-theta / omega (o)
68
70
62.5
63.0
63.5
64.0
64.5
o
50
2-theta / omega ( )
High resolution x-ray analysis
Example: strained
InxGa1-xAs on GaAs
(001) substrate
High resolution 2q/q scan near GaAs(004)
GaAs (004)
InxGa1-xAs (004)
Thickness
fringes
Lattice structure
a// film = asubstrate
0.07o
0.04o
a┴ film
>
asubstrate
(004)
InxGa1-xAs
(004)
GaAs
asubstrate
Da = sin q(substrate) - 1
a
sin q(film)
Thickness =
l
2 Dq cosq
Data: Sardela
51
Sample: Highland, Cahill, Coleman et at, UIUC
High resolution x-ray analysis
Example: strained
InxGa1-xAs on GaAs
(001) substrate
High resolution 2q/q scan near GaAs(004)
and dynamical scattering simulation
GaAs (004)
InxGa1-xAs (004)
Lattice structure
a// film = asubstrate
a┴ film
>
asubstrate
(004)
243 nm
0.76
at% In
InxGa1-xAs
(004)
GaAs
asubstrate
Takagi-Taupin dynamical scattering simulation
Data: Sardela
52
Sample: Highland, Cahill, Coleman et at, UIUC
High resolution reciprocal space mapping
• Separation of strain and mosaicity
No strain relaxation:
a// = as
• Lattice distortions within 10-5.
• Accurate lattice parameters in and out of plane
Strain relaxation:
a// ≠ as
• Strain and composition gradients
• Strain relaxation
a┴
film
a┴
• Mosaic size and rotation
film
•Misfit dislocation density
substrate
substrate
• Nanostructure dimensions,
as
• Lattice disorder and diffuse scattering.
(224)
(224)
as
substrate
substrate
film
thickness
fringes
Dq001
film
0.1 nm-1
Si1-xGex
on Si(001)
Mosaicity
(diffuse
scattering)
Dq110
0.1 nm-1
Data: Sardela et al
Si1-xGex
on Si(001)
53
High resolution reciprocal space mapping
High resolution 2q/w scan near Si(004)
Si1-xGex
Layer structure
Si substrate
Example: strained Si layer
Strained Si
on Si1-xGex / Si substrate
Strained Si (top layer, very thin)
Relaxed Si1-xGex (thick, many microns)
Si(001) substrate
Lattice structure
Strain?
Strained Si
Relaxed Si1-xGex
(virtual substrate)
(004)
Lattice distortion?
(224)
% of relaxation?
% of Ge?
Si(001) substrate
Defects?
Reciprocal lattice
(004)
(224)
Strained Si
Si substrate
Dq┴
Si1-xGex
Dq//54
High resolution reciprocal lattice map
Strained Si (top layer)
Relaxed Si1-xGex
Si(001) substrate
Map near Si(224)
(224)
Strained Si
Dq┴
Si substrate
Si1-xGex
Dq//
[001]
0.1 nm-1
Data: Sardela
Sample: Zuo, UIUC
[110]
55
High resolution reciprocal lattice map
Strained Si
e = - 0.77%
e// = 0.64 %
Si substrate
Strained Si (top layer)
Relaxed Si1-xGex
Si(001) substrate
Map near Si(224)
4.60 at% Ge
7.52 at% Ge
11.45 at% Ge
Relaxed Si1-xGex
18.70 at% Ge
100% strain relaxation
0.1 nm-1
[001]
Data: Sardela
Sample: Zuo, UIUC
[110]
56
High resolution reciprocal lattice map
Finite size
Strained Si (top layer)
Relaxed Si1-xGex
Si(001) substrate
Composition and
strain gradients
Map near Si(224)
Analyzer streaks
Coherent length
in any direction:
2p / Dqi
i = x, y, z, //, ┴
Mosaicity and
dislocations
0.1 nm-1
[001]
Data: Sardela
Sample: Zuo, UIUC
[110]
57
High resolution reciprocal lattice map
Finite size
Strained Si (top layer)
Relaxed Si1-xGex
Si(001) substrate
Composition and
strain gradients
Vertical coherent
length: 14 nm
Map near Si(224)
Analyzer streaks
Coherent length
in any direction:
2p / Dqi
i = x, y, z, //, ┴
Mosaicity and
dislocations
Misfit dislocations:
average separation : 21 nm
density: 5 x 105 cm-1
0.1 nm-1
[001]
Data: Sardela
Sample: Zuo, UIUC
[110]
58
Comparison with TEM
EDS line scan
Data: Zuo’s group, UIUC
59
The “shape” of the reciprocal lattice point
Changes in lattice
parameter
(radial direction)
224
004
Lateral sub-grain
boundaries
(along q//)
Mosaicity, curvature,
orientation
(circumferential direction)
[001]
CTR, finite
layer
thickness,
superlattice
(along q)
000
[110]
60
X-ray reflectivity methods
Bulk materials:
Near surface
region
Liquids:
Multilayered systems:
Near surface and interface information on:
Density
Porosity
Roughness
Thickness in films (ultra thin to thick)
Amorphous or crystalline materials
61
Reflectivity
Log Intensity
X-ray
source
2q
w
0.5
Detector
Sample
1.0
1.5
2.0
Angle w, q or 2q
X-ray reflectivity
Critical
angle qc
One sharp interface
Log Reflectivity R
(density re variation:
 function at interface)
q
One rough interface
Thickness fringes
Dq = l/[2*(thickness)]
(“broad”re variation
at interface)
Two interfaces
DI ~Dre
q
• Film thickness measurements: 2 – 2000 nm.
• Applicable to ultra-thin films, amorphous or crystalline materials, multilayers and liquids.
• Simulation and fitting: determination of interface roughness (rms) at each interface, roughness
correlation and film porosity.
• Very sensitive to density variations.
• Determination of critical angle, refractive index and density.
63
X-ray reflectivity analysis of thin films
Complex
multilayers
Ultra-thin
n=1
2
2
4
6
8
10
o
12
2
4
6
o
2-theta ( )
8
10
12
0.5
1.0
1.5
2.0
2.5
3.0
3.5
o
Theta ( )
sin2q
Intercept:
refractive
index h
Amorphous
PZT film
Log intensity (a.u.)
4
2-theta ( )
Data: Heitzman et al, UIUC
Metallic
multilayer
3
Log intensity (a.u.)
Log intensity (a.u.)
2.3 nm thick
polymer on Si
0
Non
crystalline
Data: Sardela, UIUC
Sample: Auoadi et al, SIU
Slope:
periodicity d
Data: Mikalsen et al, UIUC
n2
sin2q = (l2/2d)2 n2 – (h2 -1)
(modified Bragg’s law to include refractive index)
64
X-ray reflectivity data fitting in ultra-thin films
Polymer
(few nm)
1,3,5-tribromo-2-nanyloxybenze:
C15H21Br3SiO3
SiO2
(~ 100 nm)
Si substrate
SiO2
thickness
Polymer thickness
Data: Sardela
Sample: Zhang, Rogers et al, UIUC
65
X-ray reflectivity data fitting
Best fit
rms: 0.26 nm
rms: 0.45 nm
rms: 0.24 nm
Polymer
2.0 nm, 1.30 g/cm3
SiO2
98.9 nm, 2.19 g/cm3
Si substrate
Data
Best fit
66
X-ray reflectivity: summary
* Non destructive method
* Applicable to whole wafers (wafer mapping option)
* Fast method (in most cases)
* Do not depend on crystalline quality of the films
(can also be used in amorphous layers).
Quantification of:
* Layer thickness in thin films and superlattices: 1 nm ~ 1 mm (± 0.5-1%).
* Layer density and porosity (± 1-2%).
* Interface roughness: 0.1 – 10 nm (model dependent; reproducibility ~ 3%).
* Layer density gradients (variations > 2%).
* Interface roughness correlation in superlattices and multilayers.
Alternative techniques:
* Thickness: optical methods (TEM, SEM) poor contrast issues.
* Density: RBS (issues for ultra thin layers).
* Interface roughness: AFM (surface only – not buried interfaces).
67
Comparison with other techniques
X-ray analysis methods
Other techniques
Sample
preparation
and vacuum
compatibility
o No vacuum compatibility required (except
XRF on vacuum).
o “Any” sample size (depends on the
goniometer size/weight capability).
o Rough surfaces acceptable (parallel beam
configuration).
o No sample preparation required (prep
recommended for the detection of unknown
phases or elements in XRD/XRF).
o Surface analysis and electron microscopy
techniques will require vacuum compatibility
and in many cases sample preparation.
o Optical techniques will do analysis on air.
Composition
and impurity
determination
and
quantification
 ~ 0.1 w % (XRF > ppm); may require
standards.
 XRD: also phase information and % of
crystallinity.
 Data averaged over large lateral area.
 XPS: > 0.01 – 0.1 at % (may require depth
profiling).
 SIMS: > 1 ppm (requires sputtering depth
profiling).
 EDS: > 0.1 – 1 w % over small volume 1mm3.
 Little with phase information; averages over
small lateral areas (< 100 mm).
Lattice
constants
o Better than within 10-5
o TEM: estimates ~ 10-3
Thickness in
thin films
 HR-XRD or XRR: direct measurement (no
modeling for single or bi-layers).
 Requires flat interfaces.
 RBS: > 10 nm (requires modeling).
 Ellipsometry: requires modeling.
 TEM: requires visual contrast between layers.
Grain size
o Measures Crystallite Size.
o Typically ~ 1-2 nm – microns, requires
size/strain assumptions/ modeling.
o “Volume average” size.
o SEM: grain size distribution averaged over
small area.
o TEM/SEM: “number average” size.
Comparison with other techniques
X-ray analysis methods
Other techniques
Texture
o Type and distribution averaged over large
sample volume.
o EBSD: within grain sizes dimensions, better
sensitivity at the surface.
Residual Stress
 10 MPa, averaged over large sample
volume (large number of grains).
 Needs crystallinity.
 Measures strain and obtains stress from
Hooke’s law.
 Averages macro and micro stresses over
large area of a layer.
 Wafer curvature: No need for crystallinity.
Direct measurement of stress, but only
interlayer stress between film and substrate
(macrostress).
Depth
dependent
information
 Phase, grain sizes, texture and stress
“depth profiling” – requires x-ray information
depth modeling
 Surface analysis depth profiling:
compositional depth profiles.
Surface or
Interface
roughness
 XRR: interface roughness 0.01 – 5 nm,
including buried interfaces
 SPM: top surface only; rsm~ 0.01-100 nm.
Defects
 Misfit dislocations (HR-XRD).
 Point defects (diffuse scattering with
model).
 Extended defects (powder XRD with
model).
 Average over larger sample area (> mm).
 TEM: accurate identification of defects and
their densities; average over small sample
area. Sample preparation may introduce
artifacts.
Instrument cost
 Portable instruments ~ $ 60 K.
 Average well-equipped: ~ $ 200 – 300 K.
 Top of the line ~ $ 500 K (including
microdiffraction and 2D detectors).
 Surface analysis instruments > $ 500 K.
 Electron microscopes ~ $ 300 K – 1 M.
 RBS ~ $ 2 M.
 Raman, ellipsometry > $ 100 K.
X-ray analysis summary
Information
contents
• % of crystallinity and amorphous contents
• Identification and quantification of phases and mixtures
• Chemical information (if crystalline)
• Texture (type and strength) and fraction of random grains
• Grain / crystallite size (> 2 nm)
• Strain (> 10-5)
• Stress (type, direction and value)
• Relative crystallographic orientation
• Lattice constants and unit cell type
• Structure determination
• Thickness (1.5 nm – 3 mm)
• Roughness (including buried interfaces)
• Density and porosity
• Relative fraction of domains
• Defects and dislocation densities
• Mosaic tilts and sizes
•…
Detection limits
> 0.1 – 0.5 w%
Depth information
Up to 20 – 50 mm (typical)
> 10 nm and variable with glancing angle XRD
Lateral resolution
~ a few cm (typical)
10 mm (microdiffraction) – 5 cm
Angular resolution
0.1o 2q (typical powder diffraction)
0.003o 2q (high resolution methods)
Sample requirement
Mostly non destructive
Sample sizes from ~ 50 mm to many cm
No vacuum compatibility required
Solids, liquids, gels
* Semiconductors
* Coatings
* Pharmaceutical
Cements
* Metals
* Ceramics
* Geology
* Archeology
* Biosciences
* Forensics
* Medical applications
* Nanotechnology
* Polymers
* Food science
* Combustion
* Energy
*…
70
Acknowledgments
Platinum
Sponsors:
Sponsors:
© 2012 University of Illinois Board of Trustees. All rights reserved.
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