Transcript Defects in Solids - West Virginia University

Nomenclature
Usually, it is sufficient to know the
energy En(k) curves - the dispersion
relations - along the major directions.
Directions are chosen that lead aong
special symmetry points. These
points are labeled according to the
following rules:
Energy or Frequency
Direction along BZ
• Points (and lines) inside the Brillouin zone are denoted with Greek letters.
• Points on the surface of the Brillouin zone with Roman letters.
• The center of the Wigner-Seitz cell is always denoted by a G
fcc
Brillouin Zones in 3D
Note: fcc lattice in reciprocal
space is a bcc lattice
bcc
hcp
•The BZ reflects lattice symmetry
•Construction leads to primitive unit cell in rec. space
Note: bcc lattice in reciprocal
space is a fcc lattice
Brillouin
Zone of
Silicon
Symbol
Description
Γ
Center of the Brillouin zone
Simple Cubic
M
Center of an edge
R
Corner point
X
Center of a face
FCC
Middle of an edge joining
K
two hexagonal faces
L
Center of a hexagonal face C6
Middle of an edge joining a
U
hexagonal and a square face
W
Corner point
X
Center of a square face C4
BCC
What kind of
crystal structure is
Si?
Points of symmetry on the BZ are
important (e.g. determining
bandstructure).
Electrons in semiconductors are
perturbed by the potential of the
crystal, which varies across unit cell.
H
Corner point joining 4 edges
N
P
Center of a face
Corner point joining 3 edges
Learning Objectives for Diffraction
After our diffraction class you should be able to:
• Explain why diffraction occurs
• Utilize Bragg’s law to determine angles of
diffraction
• Discuss some different diffraction techniques
• (Next time) Determine the lattice type and lattice
parameters of a material given an XRD pattern and
the x-ray energy
• Alternative reference: Ch. 2 Kittel
Continuum limit:
Where the wavelength is bigger than the spacing between
atoms. Otherwise diffraction effects dominate.
Application of XRD
XRD is a nondestructive and cheap technique.
Some of the uses of x-ray diffraction are:
1.
2.
3.
4.
5.
6.
7.
Determination of the structure of crystalline
materials
Measurement of strain and small grain size
Determination of the orientation of single crystals
Measurement of layer thickness
Differentiation between crystalline and amorphous
materials
Determination of electron distribution within the
atoms, and throughout the unit cell
Determination of the texture of polygrained materials
DIFFRACTION
• Diffraction is a wave phenomenon in which the apparent
bending and spreading of waves when they meet an
obstruction is measured.
• Diffraction occurs with electromagnetic waves, such as
light and radio waves, and also in sound waves and water
waves.
• X-ray diffraction is optimally sensitive to the periodic
nature of the solid’s atomic structure.
When X-rays interact with
atoms, you get scattering
Scattering is the
emission of X-rays of
the same frequency/energy
as the incident X-rays in all
Similar to the double slit
directions (but with
experiment, this scattering will
much lower intensity)
sometimes be constructive
Will look at this again shortly
Incident beam
Zeroth Order
Physical Model for X-ray Scattering
Consider a plane wave scattering on an atom.
 incident  Ao e

ko

i ( k o  R t )

R
Atom

R'

k'
 scattered  Ao e
 
i ( k   R '  t )
Diffraction Theory
Generic incoming radiation amplitude is:
To calculate amplitude of scattered waves
at detector position, sum over
contributions of all scattering centers Pi
with scattering amplitude (form factor) f:
 In  A0 e
Pi
ri
ko
 Det   In (ri ) f (r i )eik ( Rr )
R
i
R’
source
R, R’ >> ri
 Det  A0 e
i ( k 0 R  k ' R  )
ik 0 ( R  ri )
 f (r )e
R’-ri
iri ( k 0 k ')
i
The intensity that is measured (can’t measure amplitude) is
I (K ) 

f (r )e
ir K
2
dr
K  k 'k 0
Scattering vector
The book calls K, but G is another common notation.
Diffraction Theory
K=k’-ko
k’
Pi
ri
ko
ko
R
R’
source
 In  A0 e
R’-ri
ik 0 ( R  ri )
The intensity that is measured (can’t measure amplitude) is
I (K ) 

f (r )e
ir K
2
dr
K  k 'k 0
Scattering vector
The book calls K, but G is another common notation.
The Bottom Line
I (K ) 
 f (r)e
ir K
2
dr
K  k 'k 0
If you do a whole bunch of math you can prove that the
peaks only occur when (a1, a2, a3=lattice vectors):
 
 
 
n1, n2, n3 integers
a1  K  2n1 a2  K  2n2 a3  K  2n3
Compare these relations
to the properties of
reciprocal lattice vectors:


K hkl  a1  2h


K hkl  a2  2k


K hkl  a3  2l
The Laue Condition
Replacing n1 n2 n3 with the familiar h k l, we see that these
three conditions are equivalently expressed as:




K  hb1  kb2  lb3
The Laue condition
(Max von Laue, 1911)
So, the condition for nonzero intensity is that the
scattering vector K is a translation vector of the
reciprocal lattice.
 
 
 
a1  K  2n1 a2  K  2n2 a3  K  2n3
n1, n2, n3 integers
From Laue to Bragg
The magnitude of the scattering vector K depends on the angle
between the incident wave vector and the scattered wave vector: K  k  k 0



Show vector
k
o

k'
subtraction on
k'
the board


2
d hkl
Elastic scattering requires:


2
ko  k '  k 
So from the wave vector triangle
and the Laue condition we see:

K

ko
Notice this angle is 2!


2
4
K  2k sin  
sin  
d hkl

Leaving Bragg’s law:
  2d hkl sin 
If the Bragg condition is not met, the incoming wave just moves through the
lattice and emerges on the other side of the crystal (neglecting absorption)
Bragg Equation
2d hkl sin   n
where, d is the spacing of the planes and n is the order of diffraction.
• Bragg reflection can only occur for wavelength
n  2 d
• This is why we cannot use visible light.
No diffraction occurs when the above
condition is not satisfied.
X-ray Diffraction
Typical interatomic distances in solid are of the order of 0.4 nm.
n  2 d
E  hc / 
Upon substituting this value for the wavelength into the energy
equation, we find that E is of the order of 3000 eV, which is a
typical x-ray energy.
Thus x-ray diffraction of crystals is a standard diffraction probe.
Bragg Equation:
2d sin   n
The diffracted beams (reflections) from any set
of lattice planes can only occur at particular
angles pradicted by the Bragg law.
Above are 1st, 2nd, 3rd and 4th order “reflections” from the (111) face of NaCl. Orders of
reflections are given as 111, 222, 333, 444, etc. (without parentheses!)
A single crystal specimen in a Bragg-Brentano diffractometer (θin=θout)
would produce only one family of peaks in the diffraction pattern.
2
The (110) planes would diffract at
At 20.6 °2, Bragg’s
29.3 °2; however, they are not
law fulfilled for the
properly aligned to produce a
(100) planes,
diffraction peak (the perpendicular
producing a
to those planes does not bisect the
diffraction peak. incident and diffracted beams). Only
background is observed.
The (200) planes are parallel to
the (100) planes. Therefore, they
also diffract for this crystal. Since
d200 is ½ d100, they appear at 42
°2.
THE EWALD SPHERE
Consider an arbitrary sphere
passing through the reciprocal lattice,
with the crystal arranged in the center of the sphere.
We specify two conditions:
(1) the sphere radius is 2 / - the inverse wavelength of X-ray radiation
(2) the origin of the reciprocal lattice lies on the surface of the sphere
diffracted
ray
X-rays are ON
2
2/
The diffraction spot will be observed when a
reciprocal lattice point crosses the Ewald sphere
O
20
A sphere of radius k
Surface intersects a
point in reciprocal
space and its origin is at
the tip of the incident
wavevector.
Sphere can be moved in
reciprocal lattice space
arbitrarily.
Any points which
intersect the surface of
the sphere indicate
where diffraction peaks
will be observed if the
structure factor is
nonzero (later).
The Ewald Sphere
Reciprocal Space
2
Ki
KD
02
Only a few angles
01
00
DK
10
20
(41)
The Ewald Sphere touches the
reciprocal lattice (for point 41)
 Bragg’s equation is satisfied
for 41
1. Longitudinal or θ-2θ scan
Sample moves as θ, Detector follows as 2θ
k0
k’
0
10
20
30
40
1. Longitudinal or θ-2θ scan
Sample moves on θ, Detector follows on 2θ
k0
K
k’
0
10
20
30
Reciprocal lattice
rotates by θ during
scan
40
1. Longitudinal or θ-2θ scan
Sample moves on θ, Detector follows on 2θ
k0
K
k’
2
0
10
20
30
40
1. Longitudinal or θ-2θ scan
Sample moves on θ, Detector follows on 2θ
k0
K
k’
2
0
10
20
30
40
1. Longitudinal or θ-2θ scan
Sample moves on θ, Detector follows on 2θ
k0
K
k’
2
0
10
20
30
40
1. Longitudinal or θ-2θ scan
Sample moves on θ, Detector follows on 2θ
k0
K
2k’
0
10
20
30
40
1. Longitudinal or θ-2θ scan
Sample moves on θ, Detector follows on 2θ
k0
K
k’
2
0
10
20
•Provides information about relative arrangements,
angles, and spacings between crystal planes.
30
40
Higher order diffraction peaks
http://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/ewald.php
http://www.physics.byu.edu/faculty/campbell/animations/x-ray_diffraction.html
3 COMMON X-RAY DIFFRACTION
METHODS
X-Ray Diffraction
Method
Laue
Rotating Crystal
Powder
Orientation
Single Crystal
Polychromatic Beam
Fixed Angle
Lattice constant
Single Crystal
Monochromatic Beam
Variable Angle
Lattice Parameters
Polycrystal/Powder
Monochromatic Beam
Variable Angle
Back-reflection vs. Transmission
Laue Methods
In the back-reflection method, the film
is placed between the x-ray source and
the crystal. The beams which are
diffracted backward are recorded.
Which is this?
X-rays have wide
wavelength range
(called whiteSingle
beam).
Crystal
X-Ray
Film
X-Ray
Single
Crystal
Film
The diffraction spots generally lay on:
a hyperbola
an ellipse
LAUE METHOD
The diffracted beams form arrays of spots, that
lie on curves on the film.
Each set of planes in the
crystal picks out and
diffracts
a
particular
wavelength from the white
radiation that satisfies the
Bragg law for the values of
d and θ involved.
32
Laue Pattern
The symmetry of the
spot pattern reflects the
symmetry of the crystal
when viewed along the
direction of the incident
beam.
Great for symmetry and
orientation determination
Crystal structure
determination by
Laue method?
• Although the Laue method can be used,
several wavelengths can reflect in different
orders from the same set of planes, making
structure determination difficult (use when
structure known for orientation or strain).
• Rotating crystal method overcomes this
problem. How?
ROTATING CRYSTAL
METHOD
A single crystal is mounted
with a rotation axis
perpendicular to a
monochromatic x-ray
beam.
A cylindrical film is placed
around it and the crystal is
rotated.
Sets of lattice planes will at some point make the
correct Bragg angle, and at that point a diffracted
beam will be formed.
Rotating Crystal Method
Reflected beams are located on imaginary cones.
By recording the diffraction
patterns (both angles and
intensities), one can determine
the shape and size of unit cell as
well as arrangement of atoms
inside the cell.
Film
THE POWDER METHOD
Least crystal information needed ahead of time
If a powder is used, instead of a single crystal,
then there is no need to rotate the sample,
because there will always be some crystals at an
orientation for which diffraction is permitted.
A monochromatic X-ray beam is incident on a
powdered or polycrystalline sample.
The Powder Method
• A
If the
asample
monochromatic
sample
of some
beam
tens
is

of consists
some x-ray
hundreds
of
directed
of
randomly
at aa single
orientated
crystal,
single
then
crystals
(i.e.
powdered
sample)
only one
crystals,
the
or the
two
diffracted
diffracted
beams
are
show
that
diffracted
beams
may result.
seen
to lie oncones.
the surface of
form
continuous
several
A
circle cones.
of film is used to record the
 diffraction
The cones may
point
forwards
pattern
as both
shown.
and backwards.
 Each
cone intersects the film giving
diffraction arcs.
38
Powder diffraction film
When the film is removed from the camera,
flattened and processed, it shows the diffraction
lines and the holes for the incident and transmitted
beams.
39
Useful for Phase Identification
The diffraction pattern for every phase is as unique as your
fingerprint
– Phases with the same element composition can have drastically
different diffraction patterns.
– Use the position and relative intensity of a series of peaks to match
experimental data to the reference patterns in the database
Databases such as the Powder Diffraction File (PDF)
contain dI lists for thousands of crystalline phases.
• The PDF contains over 200,000 diffraction patterns.
• Modern computer programs can help you determine
what phases are present in your sample by quickly
comparing your diffraction data to all of the patterns in
the database.
Quantitative Phase Analysis
Reference Intensity Ratio Method
I(phase a)/I(phase b) ..
• With high quality data, you can
determine how much of each phase
is present
• The ratio of peak intensities varies
linearly as a function of weight
fractions for any two phases in a
mixture
• RIR method is fast and gives semiquantitative results
• Whole pattern fitting/Rietveld
refinement is a more accurate but
more complicated analysis
60
50
40
30
20
10
0
0
0.2
0.4
0.6
X(phase a)/X(phase b)
0.8
1
Applications of Powder Diffractometry
-phase analysis (comparison to known patterns)
-unit cell determination (dhkl′s depend on lattice parameters)
-particle size estimation (line width)
-crystal structure determination (line intensities and profiles)
XRD: “Rocking” Curve Scan
K
K
ki
Sample normal
kf
“Rock” Sample
• Vary ORIENTATION of K relative to sample normal while
maintaining its magnitude.
How? “Rock” sample over a very small angular range.
• Resulting data of Intensity vs. Omega (, sample angle)
shows detailed structure of diffraction peak being
investigated. Can inform about quality of sample.
Intensity (Counts/s)
XRD: Rocking Curve Example
GaN Thin Film
(002) Reflection
16000
Compare to
literature to see
how good (some
materials naturally
easier than
others)
8000
Generally limited by
quality of substrate
0
16.995
17.195
17.395
17.595
17.795
Omega (deg)
• Rocking curve of single crystal GaN around (002)
diffraction peak showing its detailed structure.
X-Ray Reflectivity (XRR)
• A glancing, but varying, incident
angle, combined with a
matching detector angle collects
the X rays reflected from the
samples surface
• Interference fringes in the
reflected signal can be used to
determine:
– thickness of thin film layers
– density and composition of thin
film layers
– roughness of films and interfaces
X-ray reflectivity measurement
Calculation of the electron density, thickness and
interface roughness for each particular layer
r
6
10
Mo
Edge of TER
0.68
t [Å] s [Å]
19.6
5.8
5
Intensity (a.u.)
10
Kiessig oscillations (fringes)
4
10
Mo
0.93 236.5 34.0
3
10
2
10
1
10
0
10
W
1.09
1.00
Si
1.00
Mo
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
14.1
5.0
2.7
2.7
2.8
o
Diffraction angle ( 2)
The surface must be smooth
(mirror-like)
Lots of extra slides
• There is a lot of useful information on
diffraction. Following are some related slides
that I have used or considered using in the
past.
• A whole course could be tough focusing on
diffraction so I can’t cover everything here.
XRD: Reciprocal-Space Map
AlN
GaN(002)

/2
• Vary Orientation and Magnitude of Dk.
• Diffraction-Space map of GaN film on AlN buffer
shows peaks of each film.
Preferred Orientation
(texture)
Diffracting crystallites
• Preferred orientation of crystallites can create a
systematic variation in diffraction peak intensities
– can qualitatively analyze using a 1D diffraction pattern
– a pole figure maps the intensity of a single peak as a
function of tilt and rotation of the sample
• this can be used to quantify the texture
10.0
00-004-0784> Gold - Au
(111)
Intensity(Counts)
8.0
(311)
(200)
6.0
(220)
4.0
(222)
2.0
(400)
x10
3
40
50
60
70
Two-Theta (deg)
80
90
100
The X-ray Shutter is the most important
safety device on a diffractometer
• X-rays exit the tube through X-ray
transparent Be windows.
H2O In
H2O Out
XRAYS
Be
window
• X-Ray safety shutters contain the
beam so that you may work in the
diffractometer without being
exposed to the X-rays.
Cu
ANODE
Be
window
Primary
Shutter
eXRAYS
Secondary
Shutter
FILAMENT
(cathode)
Solenoid
metal
glass
(vacuum)
• Being aware of the status of the
shutters is the most important
factor in working safely with X
rays.
(vacuum)
AC CURRENT
SAFETY SHUTTERS
Diffraction Methods
• Any particle will scatter and create diffraction
pattern
• Beams are selected by experimentalists depending
on sensitivity
– X-rays not sensitive to low Z elements, but neutrons are
– Electrons sensitive to surface structure if energy is low
– Atoms (e.g., helium) sensitive to surface only
• For inelastic scattering, momentum conservation is
important
Electron
X-Ray
Neutron
λ = 1A°
λ = 1A°
λ = 2A°
E ~ 104 eV
E ~ 0.08 eV
E ~ 150 eV
interact with electron
Penetrating
interact with nuclei
Highly Penetrating
interact with electron
Less Penetrating
Electron Diffraction
(Covered in Chapter 18)
If low electron energies are used, the penetration depth
will be very small (only about 50 A°), and the beam will
be reflected from the surface. Consequently, electron
diffraction is a useful technique for surface structure
studies.
Electrons are scattered strongly in air, so diffraction
experiment must be carried out in a high vacuum. This
brings complication and it is expensive as well.
54
Electron Diffraction
Electron diffraction has also been used in the analysis of crystal
structure. The electron, like the neutron, possesses wave properties;
 2k 2
h2
E

 40eV
2
2me 2me
  2A
0
Electrons are charged particles and interact strongly with all
atoms. So electrons with an energy of a few eV would be completely
absorbed by the specimen. In order that an electron beam can
penetrate into a specimen , it necessitas a beam of very high energy
(50 keV to 1MeV) as well as the specimen must be thin (100-1000 nm)
55