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Lecture Module #14
Introduction to Electron Microscopy
An overview
What are electron microscopes?
• Scientific instruments that use a focused beam of
electrons to examine objects on a very fine scale.
What is electron microscopy?
• Electron microscopy is the science and technology of
using an electron beam to form a magnified image.
• Advantages:
– The use of electrons rather than light provides a nearly 1000
fold increase in resolving power (i.e., ability to focus fine
details) over light.
• Disadvantages:
– High cost
– Time commitment
– Small areas of analysis
Magnification
• Refers to how large an object can be made (and still
resolved).
• Advantages:
– You get higher magnification with electron microscopes
techniques than you can with light.
– Smaller wavelength Æ higher resolving power.
object
of
size
image
of
size
ion
magnificat =
Resolution is:
• The closest distance between two points that can
clearly be resolved as separate entities through the
microscope.
d1
ro
Intensity
Distance
Consider light as
EM energy
transmitted as a
wave motion.
We can consider
light as a series of
ripples impinging
upon an obstacle.
Abbe relationship
NA
n
d
ro
λ
α
λ 61
.
0
sin
61
.
0
2
1
=
=
=
λ = wavelength of illuminant
α = semi-angle
n = index of refraction
NA = numerical aperature
Resolution is defined by:
↓
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
Depth of Field
• A measure of how much of the object that we are
looking at remains in focus at the same time.
• Advantages:
– You get higher depth of field with (many) electron
microscopy techniques than you do with light.
– WHY?
Important Terms
• Depth of Field
– Height above and below the plane of focus that an image
remains sharp.
– DOF is a function of magnification, α, and probe size
Plane of focused image
α
BEAM
Scan
Region of image
in focus
DOF
SEM
SEM
TEM
TEM
EPMA
EPMA
Other instruments we have or are getting
• DualBeam Focused Ion Beam (DB-FIB)
– Nano-machining platform
• X-ray photoelectron spectrometer / Auger (XPS/Auger)
• ???
What do we get out of electron
microscopes?
• Topography
– Surface features of an object. “How it looks.”
• Morphology
– Size and shape of particles making up object.
• Composition
– Relative amount of elements and compounds making up the
object.
• Structure
– Crystallography. How atoms are arranged in the object
Comparison of Selected Characteristics of Light
and Electron Microscopes
FEATURE Light Microscope SEM TEM
Uses Surface morphology
and sections (1-40 µm)
Surface morphology Sections (40-150 nm)
or small particles on
thin membranes
Source of
Illumination
Visible light High-speed electrons High-speed electrons
Best resolution 200 nm 3-6 nm 0.2 nm
Magnification range 10-1,000× 20-150,000× 500-500,000×
Depth of field 0.002-0.05 nm
(NA=1.5)
0.003-1 mm 0.004-0.006 mm
(NA=10-3
)
Lens type Glass Electromagnetic Electromagnetic
Image ray-
formation spot
On eye by lens On CRT by scanning
device
On phosphorescent
screen by lens
Information
generated
Phases
Reflectivity
Topography
Composition
Crystal orientation
Crystal structure
Crystal orientation
Defects
Composition
Limiting Factor Wavelength of light Brighness,
signal/noise ratio,
emission volume
Lens quality
Adapted from: Scanning and Transmission Electron Microscopy, S. L. Flegler, J.W. Heckman Jr., K.L. Klomparens,
Oxford University Press, New York, 1993.
All of this information is related to
properties.
Mechanical, Physical, Thermal, Optical, Electrical, etc…
structure
properties
processing
performance
History of electron microscopes
• Developed due to limitations of light optical
microscopes (LOMs)
– LOM: ~1000x magnification; 0.2 µm (200 nm) resolution
• Transmission Electron Microscope (TEM) was
developed first.
– M. Knoll and E. Ruska, 1931
– Patterned “exactly” like a LOM. Uses electrons rather than
light.
• Scanning Electron Microscope (SEM)
– 1942
Goodhew, Humphreys & Beanland
Electron Microscopy and Microanalysis, 3rd
Edition
Taylor & Francis, London, 2001 70X
300X
1400X
2800X
Electron microscopy
provides significant
advantages over light
optical microscopy in
terms of resolution
and depth of field
How do electron microscopes work?
• Form a stream of electrons in the electron source
(thermionic emission) and accelerate them towards
the specimen using a positive electrical potential.
• Use apertures and magnetic lenses to focus the
stream into a thin focused monochromatic beam
• Focus the beam onto the sample using another
magnetic lens.
• Interactions occur inside the irradiated area of the
sample. Collect results of interactions in a suitable
detector and transform them into an image (or
whatever you are interested in).
The Electron Gun
• The electron gun provides an intense beam of high
energy electrons.
• There are two main types of gun. The thermionic
gun, which is the most commonly used, and the field
emission gun.
• Electrons are generated and emitted from a filament
by thermionic emission (W or LaB6) or from a sharp
tip by field emission (single crystal of W).
• The electrons are accelerated through a potential
difference.
Electron Source
(i.e., electron gun)
Thermionic electron gun
most widely used (~$70K-$150K)
Electrons are emitted
from a heated
filament and
accelerated towards
an anode
Wehnelt cap
(negative potential)
W Filament
Anode Plate
(positive potential)
Electron
Beam
Space charge
Crossover
–
10-1000 kV
+
Ground
Bias
resistor
Wehnelt cap
(negative potential)
W Filament
Anode Plate
(positive potential)
Electron
Beam
Space charge
Crossover
–
10-1000 kV
+
Ground
Bias
resistor
Filament
Made from a high Tmp material
with a low work function (φ) in
order to emit as many electrons
as possible. The work function
is the energy needed by an
electron to overcome the barrier
that prevents it from leaking out
of the atom. φW = 4.5 eV, φLaB6 =
3.0 eV.
Wehnelt
A biased grid with a potential that is
a few hundred volts different than
the filament (cathode). This helps
to accelerate the electrons and
causes their paths to cross over.
Crossover
The effective source of
illumination for the
microscope. The size
is critical for high
resolution
applications.
Anode
Positively charged metal plate
at earth potential with a hole in
it. It accelerates the electron
beam to the high tension
potential.
Operation of thermionic gun
• Apply a positive electrical potential to the anode
• Heat the cathode (filament) until a stream of electrons is
produced
– >2700 K for W
• Apply a negative electric potential to the Wehnelt
– electrons are repelled by the Wehnelt towards the optic axis
• Electrons accumulate within the region between the filament tip
and the Wehnelt. This is known as the space charge.
• Electrons near the hole exit the gun and move down the column
to the target (in this case the sample) for imaging.
Other Types of Electron Sources
(i.e., electron guns)
Lanthanum hexaboride (LaB6)
Higher brightness than W. More $$$ (~$500K)
Field emission source
Highest brightness. Even more $$$ (~$750K)
ADVANTAGES
Provide higher current density → Higher brightness
http://www.matter.org.uk/tem/electron_gun/electron_sources.htm
Brightness, B, is the beam
current density per unit solid
angle.
B = jcπβ2
jc = current density
β = convergence angle
The function of the electron gun is:
To provide an intense beam of high
energy electrons
There are two main types of gun. The
thermionic gun, which is the most
commonly used, and the field emission
gun.
• This is what makes electron microscopy possible.
• Electrons strike the sample leading to a variety of reactions.
• Use results of reactions to form image or generate other
information.
Beam-Specimen Interaction
Incident high kV beam of
electrons
Secondary e-
(SE)
Characteristic
x-rays
Visible light
Backscattered
e- (BSE)
Auger e-
Inelastically
scattered e-
Elastically
scattered e-
Bremsstrahlung
x-rays
(noise)
Direct
(transmitted)
beam
Absorbed e- e- - hole pairs
Bulk (SEM)
Foil (TEM)
Interaction Volume
Interaction Volume
• Represents the region penetrated by electrons
– Signals must escape the sample to be detected
Secondary electrons
Backscattered electrons
X-rays
Incident electron beam
~1 nm – Auger electrons
~5–50 nm
Secondary electrons
~300 nm
Backscattered
electrons
~50 nm
TEM
specimen
thickness
~1.5 µm
X-rays
~1.0 µm
X-ray
resolution
Z = 29 (Cu), Accelerating voltage = 20kV
INCIDENT BEAM
Different signals come from
different depths in the target
(sample)
• X-rays:
– most escape
– sampling volume ≈ interaction volume
• Backscattered electrons:
– originate further from the incident beam (some fraction of a µm)
• Secondary electrons:
– escape from a region slightly larger than the incident beam
(several µm)
– yield the best spatial resolution
Interaction Volume
Backscattered electrons (BSE)
• Formation
– Caused when incident electrons collide with an atom in a specimen
that is nearly normal to the path of the incident beam.
– Incident electron is scattered backward (“reflected”).
• Use
– Imaging and diffraction analysis in the SEM.
– Production varies with atomic number (Z).
– Higher Z elements appear brighter than lower Z elements.
– Differentiate parts of specimen having different atomic number
Backscattered electrons are not as numerous as others. However,
they generally carry higher energies than other types of electrons.
Secondary Electrons (SE)
• Formation
– Caused when an incident electron “knocks” and inner shell
electron (e.g., k-shell) out of its site.
– This causes a slight energy loss and path change in the
incident electron and ionization of the electron in the
specimen.
– The ionized electron leaves the atom with a small kinetic
energy (~5 eV)
• Use
– IMAGING!
– Production is related to topography. Due to low energy, only
SE near the surface can exit the sample.
– Any change in topography that is larger than the sampling
depth will change the yield of SE.
More abundant than other types of electrons. They are electrons
that escape the specimen with energies below ~50eV
Auger Electrons (AE)
• Formation
– De-energization of the atom after a secondary electron is
produced.
– During SE production, an inner shell electron is emitted from
the atom leaving a vacancy.
– Higher energy electrons from the same atom can fall into the
lower energy hole. This creates an energy surplus in the
atom which is corrected by emission of an outer shell (low
energy) electron
• Use
– AE have characteristic energies that are unique to each
element from which they are emitted.
– Collect and sort AE according to energy to determine
composition.
– AE have very low energy and are emitted from near surface
regions.
X-rays
• Formation
– Same as AE. Difference is that the electron that fills the
inner shell emits energy to balance the total energy of the
atom.
• Use
– X-rays will have characteristic energies that are unique to
the element(s) from which it originated.
– Collect and sort signals according to energy to yield
compositional information.
– Energy Dispersive X-ray Spectroscopy (EDS)
Foundation of XPS (X-ray photoelectron spectroscopy). XPS can
be used to determine the “state” of an atom and to identify chemical
compounds.
Transmitted electrons
• Can be used to determine:
– thickness
– crystallographic orientation
– atomic arrangements
– phases present
– etc.
• Foundation for Transmission Electron Microscopy (TEM)
Specimen Interaction Volume
• Volume where electron beam - specimen interactions
occur.
• Depends upon:
– Z of material being examined
• higher Z materials absorb more electrons and have smaller
interaction volume
– Accelerating Voltage
• higher voltages penetrate further into the specimen and
generate larger interaction volumes
– Angle of incidence of electron beam
• larger angle leads to a smaller interaction volume
~1 nm – Auger electrons
~5–50 nm
Secondary electrons
~300 nm
Backscattered
electrons
~50 nm
TEM
specimen
thickness
~1.5 µm
X-rays
???
X-ray resolution
Z = 29 (Cu), Accelerating voltage = 20kV
INCIDENT BEAM
10 Å - 30 Å – Auger electrons
~100 Å
Secondary electrons
<1µm - 2µm
Backscattered
electrons
~5 µm
X-rays
~1.0 µm
X-ray resolution
Z = 28 (Ni), Accelerating voltage = 20kV
INCIDENT BEAM
~50 nm
TEM
specimen
thickness
• Interaction volume is larger for materials that have
lower atomic numbers and for higher incident beam
energies!
~50 nm
Increasing atomic number (Z)
Increasing incident energy (E0)
Z1
Z2
Z3
Z1 < Z2 < Z3
“Instruments of the trade”
• Primary Instruments
– Transmission Electron Microscope (TEM)
– Scanning Electron Microscope (SEM)
• Variants
– Electron Probe Microanalyzer (EPMA)
– Scanning Transmission Electron Microscope (STEM)
– Environmental SEM (ESEM)
– High Resolution TEM (HRTEM)
– High Voltage TEM (HVTEM)
– etc...
TEM
• Patterned after transmission light optical microscopes
• Yield Following Information:
– Morphology
• Size shape and arrangement of particles, precipitates, etc.
– Crystallographic information
• Atomic arrangement
– Compositional Information
• If proper detector is present
The TEM is analogous to the transmission LOM
Source
Condenser
system
Stage
Objective
lens
Final imaging
system
Observation and
recording system
x
y
z
+ Tilt & rotation
Vacuum
column
Apertures
x
y
z
1 in 1020 electrons
are collected
<1 in 106 photons are
collected
TEM LOM
Phosphor
screen; film;
digital recording
system
Eyepiece or TV
screen; film;
digital recording
system
Light source
Electron gun
Components of the TEM
• Source – filament plus anode plates with
applied accelerating voltage.
• Condenser Lenses – electromagnetic lenses
adjusted by lens currents not position.
• Specimen Stage – allows translations and tilts.
• Objective Lens – usually <50X.
• Imaging System – multiple electromagnetic
lenses below the objective: set magnification,
focal plane (image vs. diffraction pattern).
• Observation – fluorescent screen, plate film,
CCD camera.
x
y
z
+ Tilt & rotation
Vacuum
column
1 in 1020 electrons
are collected
Phosphor
screen; film;
digital recording
system
Electron gun
Adapted from
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
Condensor Lenses
ELECTROMAGNETIC
lenses focus the
electron beam to as
small a spot as is
possible. They are
equivalent to convex
lenses in optical lens
systems.
Light Source
OPTICAL LENS
Cu coils
Soft Fe
pole
pieces
N
S
N
S
Electron Source
MAGNETIC LENS
Adapted from
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
Electron Lenses
• Use electrostatic or electromagnetic fields to focus
beams of charged electrons.
• ELECTROMAGNETIC – Most lenses are of this type.
Consist of Cu wire coils around soft Fe cores.
Sometimes an Fe pole-piece is used to “shape” the
field.
• ELECTROSTATIC – Unusual. Only common
example is the Wehnelt aperture in the electron gun.
Electrons in Magnetic Fields
A. Electrons moving through a perpendicular magnetic field experience
perpendicular forces.
B. Electrons moving parallel to a magnetic field are unaffected.
C. Electrons moving nearly parallel to a magnetic field take a helical
path around the magnetic field direction (i.e., the microscope axis).
electron
Magnetic
field
Force on
electron is
out of plane
No Force
Electrons
follow a
helical path
unaffected
Nearly
aligned
electron
A B C
Adapted from
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
Trajectories in Electromagnetic Lenses
• When you adjust the magnification (and the focal length),
you modify the lens strength by adjusting the current in the
electromagnetic lens coils.
• Since the magnetic field is changed,
so are the helical trajectories.
• This ultimately leads to image
rotation which must be corrected
for or calibrated.
Rotated
image
Object
Cu coils
Soft Fe pole
pieces
N
S
N
S
Image is inverted
and rotated
Electron Source
MAGNETIC LENS
Beam Tilting and Translation
• The electron beam can be positioned for fine
measurements (spot modes) or scanning (SEM, STEM)
BEAM TRANSLATION
BEAM TILTING
APPARENT
ORIGIN
APPARENT
ORIGIN
Upper scan
coil
Lower scan
coil
Condenser Alignment
• In TEM it is important to center the condenser aperture about
the optical axis.
• If the aperture is off center, the beam is displaced away from
the axis when the condenser lens is focused.
• You can try condenser alignment at:
http://www.matter.org.uk/tem/condenser_alignment.htm
Condenser Astigmatism
• Astigmatism in the condenser lenses distorts the beam to an
elliptical shape either side of focus and prevents the beam
being fully focused. It is corrected by applying two
orthogonal correction fields in the x and y directions.
• You can try stigmator adjustment at:
http://www.matter.org.uk/tem/condenser_astigmatism.htm
Objective Aperture
• Depending upon where you position the objective aperture
and whether or not you tilt the beam, a bright field image, a
dark field (i.e., scattered) image, or a diffraction pattern can
be collected.
• You can look at this in more detail at:
http://www.matter.org.uk/tem/dark_field.htm
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) AND http://www.matter.org.uk/tem/stem_images.htm
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
BF TEM Micrograph and SADP
Grain structure
Crystallographic information
[111] zone axis
NiAl-1.0Hf
250 nm
1hr
NiAl-1.0Hf
BF TEM Micrographs
Precipitate distribution
Defect structures
4hr
NiAl-1.0Hf
100 nm
β′
Defect Structures
Dislocations imaged in NiAl-0.5Zr single crystals deformed at elevated temperatures.
Nano-scale X-ray Analysis
β′ -NiAl
EDX spectrum
β′ -precipitate
β′
HAADF (Z-contrast imaging) analysis
2
1
Ni2AlHf
Hf rich region
• Hf segregates to GB’s in as-deposited & annealed coatings.
• Hf precipitates as β´-Ni2AlHf phase after annealing.
NiAl-1.0Hf
NiAl-1.0Hf
As-deposited Annealed 1000°C/1h
20 nm
200 nm
How does a TEM work?
• Pass a focussed beam of electrons through a thin foil
• As beam passes through sample, it is scattered
• Project the transmitted (scattered) beam onto a
phosphor screen to form an enlarged image
• Modes:
– Bright Field/Dark Field modes for visualization of structure
and defects
– Selected Area Diffraction/Convergent Beam Diffraction
for crystallographic information
Electron Beam
Thin Foil
< 500 nm thick
Electron Gun
Anode Plate
First Condenser Lens (C1)
Second Condenser Lens (C2)
Condenser Aperture
Sample
Objective Lens
Objective Aperture
Selected Area Aperture
First Intermediate Lens
Second Intermediate Lens
Projector Lens
Phosphor Screen
Used to
magnify image.
Schematic
Representation
of optics in a
TEM
Technical Details
• Produce a stream of monochromatic electrons in the electron gun
• Focus the stream into a small coherent beam using C1 and C2
– C1 determines the “spot size” (i.e., size of electron probe)
– C2 changes intensity or brightness
• Use condenser aperture to restrict the beam
• Part of the beam is transmitted through the sample
• Focus transmitted portion using the objective lens to form an image
• Objective and selected area apertures are used to restrict the beam further
– allows examination of diffraction from specific atoms, crystals, features…
– SAD, CBD
• Enlarge image with intermediate and projector lenses
SEM
• Patterned after reflecting light optical microscopes
• Yield Following Information:
– Topography
• Surface features of an object. Detectable features limited to a
few nanometers depth.
– Morphology
• Size shape and arrangement of particles, precipitates, etc
– Compositional Information
• Elements and compounds the sample is composed of
– Crystallographic information
• Possible using new techniques (OIM/BKD)
x
y
z
Vacuum
column
x
y
z
TEM SEM
Electron gun
Condenser system
Stage
Objective lens
Final imaging system
Observation
and recording
system
Probe lens
Stage
Scan coils
Time
base
Signal
amplifier
CRT
Condenser lens
Also tilts and rotates
Also tilts and rotates
In the SEM you use secondary signals to acquire images.
Electron gun
Detector and
processing
system
Components of an SEM
• Source:
– same as TEM but lower V
• Condenser:
– same as TEM
• Scan Coils:
– raster the probe
• Probe Lens:
– lens that forms a spot at the
specimen surface
• Detector & Processing System:
– collects signals such as X-rays and
electrons as a function of time and
position.
– Provides digital images for real-time
viewing, processing, and storage.
x
y
z
Probe lens
Stage
Scan coils
Time
base
Signal
amplifier
CRT
Condenser lens
Also tilts and rotates
Electron gun
Detector and
processing
system
Schematic
Representation
of the optics in
an SEM
Electron Gun
Anode Plate
First Condenser Lens (C1)
Second Condenser Lens (C2)
Condenser Aperture
Sample
Objective (probe) Lens
Objective Aperture
Scan Coils
Detector
SEM: Technical Details
• Produce a stream of monochromatic electrons in the electron gun
• Focus the stream using the first condenser lens
– Coarse probe current knob
• The beam is constricted by the condenser aperture (eliminates high-angle electrons)
• Second condenser lens is used to form electrons into a thin, tight, coherent beam.
– Use fine probe current knob
• Use objective aperture to limit beam (i.e., eliminate high-angle electrons)
• Scan coils raster the beam across the sample, dwelling on the points for a
predetermined period of time (selected using scan speed)
• Final objective lens focuses beam on desired region.
• When beam strikes the sample, interactions occur. We detect what comes out of the
sample.
Secondary electron mode
Could get
additional
information in
backscattered or
mixed mode
10 µm
Substrate IDZ Coating
γ′
γ γ+γ′
γ+γ′
40
50
60
70
80
0
10
20
30
40
0
10
20
30
40
-15 -10 -5 0 5 10 15 20 25
0.0
0.2
0.4
0.6
0.8
1.0
γ'
Substrate γ + γ'
Composition
(at%)
Ni-AD
Ni-Annealed
Ni
Al-AD
Al-Annealed
Al
Pt-AD
Pt-Annealed
Pt
Hf-AD
Hf-Annealed
Hf
Distance (µm)
Figure 4.8. Composition profiles of the major elements
measured by WDS in an EPMA.
X-ray Analysis in the SEM
EDS or WDS
Elemental diffusion profile NiAlPtHf coatings on CMSX-4 substrates
for a temperature of 1150°C and times from (a) 20 hrs and (b) 100 hrs.
Al O
Ni
Pt
Hf
Al Ni
O
Pt
Hf
Al2O3 + ???
???
• Change magnification by changing the scan area
• Focus by changing the objective lens current
• The probe current/spot size are controlled by the
condenser lens current
Useful things to remember about
SEM operation
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
Microanalysis in Electron Microscopy
• Characteristic X-rays are always generated by
interactions between the incident electron beam and
the sample. They constitute a fingerprint of the local
chemistry.
• Collect X-ray signal to determine local chemistry
• Common Techniques:
– Wavelength-Dispersive Spectrometry/Spectroscopy (WDS)
– Energy-Dispersive Spectrometry/Spectroscopy (EDS)
WDS
• Yields best discrimination of emitted X-ray signal
• Use a series of bent crystals to cover the range of
wavelengths of interest
• Scan wavelengths within each range by rotating the
crystal and moving the detector while keeping the
position of the crystal fixed.
• X-rays are collected from the sample at a fixed angle.
The angle at the collecting crystal will vary with 2θ
and the diameter of the focusing circle will change
Specimen
X-ray Detector
Receiving slits
Bent crystal
Radius = 2R
R
Focusing Circle
WDS
WDS continued
• All peaks are scanned sequentially. You cannot
record more than one characteristic line at once
unless there are multiple spectrometers available.
• EPMA!
• Data is collected sequentially. Takes more time but
the results are much more precise than EDS.
Resolution of M lines in a WDS spectrum of a superalloy
(Oxford Instruments)
EDS vs. WDS
• Pulse height is recorded by the detector. It is related
to the energy of the photon responsible for the pulse.
• Solid-state detectors are generally used.
• EDS is faster than WDS
• Problems with EDS:
– Poor discrimination or energy resolution. WDS systems are
much better, particularly when characteristic lines from
different elements overlap.
– Need a windowless or thin window detector to pick up light
elements.
– WDS is more quantitative
The BSE sampling volume is large which limits resolution
Multiple elements of BSE
detector
“Overlapping shadows”
Reduced resolution
SE image
Single segment of BSE
detector
Can use BSE signal in conjunction with SE signal to yield enhanced topographical
information Images
• Topography and atomic number influence signal.
• Backscatter coefficient:
– η = -0.254 + 0.016Z - 1.86x10-4Z2 + 8.3x10-7Z3
• Atomic number contrast:
– C = (η1 - η2)/ η1
TOPOGRAPHIC
CONTRAST
TOPOGRAPHIC
CONTRAST
Signals from 2
detectors
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)

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Introduction to Electron Microscopyy.pdf

  • 1. Lecture Module #14 Introduction to Electron Microscopy An overview What are electron microscopes? • Scientific instruments that use a focused beam of electrons to examine objects on a very fine scale. What is electron microscopy? • Electron microscopy is the science and technology of using an electron beam to form a magnified image. • Advantages: – The use of electrons rather than light provides a nearly 1000 fold increase in resolving power (i.e., ability to focus fine details) over light. • Disadvantages: – High cost – Time commitment – Small areas of analysis Magnification • Refers to how large an object can be made (and still resolved). • Advantages: – You get higher magnification with electron microscopes techniques than you can with light. – Smaller wavelength Æ higher resolving power. object of size image of size ion magnificat =
  • 2. Resolution is: • The closest distance between two points that can clearly be resolved as separate entities through the microscope. d1 ro Intensity Distance Consider light as EM energy transmitted as a wave motion. We can consider light as a series of ripples impinging upon an obstacle. Abbe relationship NA n d ro λ α λ 61 . 0 sin 61 . 0 2 1 = = = λ = wavelength of illuminant α = semi-angle n = index of refraction NA = numerical aperature Resolution is defined by: ↓ B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) Depth of Field • A measure of how much of the object that we are looking at remains in focus at the same time. • Advantages: – You get higher depth of field with (many) electron microscopy techniques than you do with light. – WHY? Important Terms • Depth of Field – Height above and below the plane of focus that an image remains sharp. – DOF is a function of magnification, α, and probe size Plane of focused image α BEAM Scan Region of image in focus DOF
  • 3. SEM SEM TEM TEM EPMA EPMA Other instruments we have or are getting • DualBeam Focused Ion Beam (DB-FIB) – Nano-machining platform • X-ray photoelectron spectrometer / Auger (XPS/Auger) • ??? What do we get out of electron microscopes? • Topography – Surface features of an object. “How it looks.” • Morphology – Size and shape of particles making up object. • Composition – Relative amount of elements and compounds making up the object. • Structure – Crystallography. How atoms are arranged in the object Comparison of Selected Characteristics of Light and Electron Microscopes FEATURE Light Microscope SEM TEM Uses Surface morphology and sections (1-40 µm) Surface morphology Sections (40-150 nm) or small particles on thin membranes Source of Illumination Visible light High-speed electrons High-speed electrons Best resolution 200 nm 3-6 nm 0.2 nm Magnification range 10-1,000× 20-150,000× 500-500,000× Depth of field 0.002-0.05 nm (NA=1.5) 0.003-1 mm 0.004-0.006 mm (NA=10-3 ) Lens type Glass Electromagnetic Electromagnetic Image ray- formation spot On eye by lens On CRT by scanning device On phosphorescent screen by lens Information generated Phases Reflectivity Topography Composition Crystal orientation Crystal structure Crystal orientation Defects Composition Limiting Factor Wavelength of light Brighness, signal/noise ratio, emission volume Lens quality Adapted from: Scanning and Transmission Electron Microscopy, S. L. Flegler, J.W. Heckman Jr., K.L. Klomparens, Oxford University Press, New York, 1993.
  • 4. All of this information is related to properties. Mechanical, Physical, Thermal, Optical, Electrical, etc… structure properties processing performance History of electron microscopes • Developed due to limitations of light optical microscopes (LOMs) – LOM: ~1000x magnification; 0.2 µm (200 nm) resolution • Transmission Electron Microscope (TEM) was developed first. – M. Knoll and E. Ruska, 1931 – Patterned “exactly” like a LOM. Uses electrons rather than light. • Scanning Electron Microscope (SEM) – 1942 Goodhew, Humphreys & Beanland Electron Microscopy and Microanalysis, 3rd Edition Taylor & Francis, London, 2001 70X 300X 1400X 2800X Electron microscopy provides significant advantages over light optical microscopy in terms of resolution and depth of field How do electron microscopes work? • Form a stream of electrons in the electron source (thermionic emission) and accelerate them towards the specimen using a positive electrical potential. • Use apertures and magnetic lenses to focus the stream into a thin focused monochromatic beam • Focus the beam onto the sample using another magnetic lens. • Interactions occur inside the irradiated area of the sample. Collect results of interactions in a suitable detector and transform them into an image (or whatever you are interested in).
  • 5. The Electron Gun • The electron gun provides an intense beam of high energy electrons. • There are two main types of gun. The thermionic gun, which is the most commonly used, and the field emission gun. • Electrons are generated and emitted from a filament by thermionic emission (W or LaB6) or from a sharp tip by field emission (single crystal of W). • The electrons are accelerated through a potential difference. Electron Source (i.e., electron gun) Thermionic electron gun most widely used (~$70K-$150K) Electrons are emitted from a heated filament and accelerated towards an anode Wehnelt cap (negative potential) W Filament Anode Plate (positive potential) Electron Beam Space charge Crossover – 10-1000 kV + Ground Bias resistor Wehnelt cap (negative potential) W Filament Anode Plate (positive potential) Electron Beam Space charge Crossover – 10-1000 kV + Ground Bias resistor Filament Made from a high Tmp material with a low work function (φ) in order to emit as many electrons as possible. The work function is the energy needed by an electron to overcome the barrier that prevents it from leaking out of the atom. φW = 4.5 eV, φLaB6 = 3.0 eV. Wehnelt A biased grid with a potential that is a few hundred volts different than the filament (cathode). This helps to accelerate the electrons and causes their paths to cross over. Crossover The effective source of illumination for the microscope. The size is critical for high resolution applications. Anode Positively charged metal plate at earth potential with a hole in it. It accelerates the electron beam to the high tension potential. Operation of thermionic gun • Apply a positive electrical potential to the anode • Heat the cathode (filament) until a stream of electrons is produced – >2700 K for W • Apply a negative electric potential to the Wehnelt – electrons are repelled by the Wehnelt towards the optic axis • Electrons accumulate within the region between the filament tip and the Wehnelt. This is known as the space charge. • Electrons near the hole exit the gun and move down the column to the target (in this case the sample) for imaging.
  • 6. Other Types of Electron Sources (i.e., electron guns) Lanthanum hexaboride (LaB6) Higher brightness than W. More $$$ (~$500K) Field emission source Highest brightness. Even more $$$ (~$750K) ADVANTAGES Provide higher current density → Higher brightness http://www.matter.org.uk/tem/electron_gun/electron_sources.htm Brightness, B, is the beam current density per unit solid angle. B = jcπβ2 jc = current density β = convergence angle The function of the electron gun is: To provide an intense beam of high energy electrons There are two main types of gun. The thermionic gun, which is the most commonly used, and the field emission gun. • This is what makes electron microscopy possible. • Electrons strike the sample leading to a variety of reactions. • Use results of reactions to form image or generate other information. Beam-Specimen Interaction Incident high kV beam of electrons Secondary e- (SE) Characteristic x-rays Visible light Backscattered e- (BSE) Auger e- Inelastically scattered e- Elastically scattered e- Bremsstrahlung x-rays (noise) Direct (transmitted) beam Absorbed e- e- - hole pairs Bulk (SEM) Foil (TEM) Interaction Volume Interaction Volume • Represents the region penetrated by electrons – Signals must escape the sample to be detected Secondary electrons Backscattered electrons X-rays Incident electron beam
  • 7. ~1 nm – Auger electrons ~5–50 nm Secondary electrons ~300 nm Backscattered electrons ~50 nm TEM specimen thickness ~1.5 µm X-rays ~1.0 µm X-ray resolution Z = 29 (Cu), Accelerating voltage = 20kV INCIDENT BEAM Different signals come from different depths in the target (sample) • X-rays: – most escape – sampling volume ≈ interaction volume • Backscattered electrons: – originate further from the incident beam (some fraction of a µm) • Secondary electrons: – escape from a region slightly larger than the incident beam (several µm) – yield the best spatial resolution Interaction Volume Backscattered electrons (BSE) • Formation – Caused when incident electrons collide with an atom in a specimen that is nearly normal to the path of the incident beam. – Incident electron is scattered backward (“reflected”). • Use – Imaging and diffraction analysis in the SEM. – Production varies with atomic number (Z). – Higher Z elements appear brighter than lower Z elements. – Differentiate parts of specimen having different atomic number Backscattered electrons are not as numerous as others. However, they generally carry higher energies than other types of electrons. Secondary Electrons (SE) • Formation – Caused when an incident electron “knocks” and inner shell electron (e.g., k-shell) out of its site. – This causes a slight energy loss and path change in the incident electron and ionization of the electron in the specimen. – The ionized electron leaves the atom with a small kinetic energy (~5 eV) • Use – IMAGING! – Production is related to topography. Due to low energy, only SE near the surface can exit the sample. – Any change in topography that is larger than the sampling depth will change the yield of SE. More abundant than other types of electrons. They are electrons that escape the specimen with energies below ~50eV
  • 8. Auger Electrons (AE) • Formation – De-energization of the atom after a secondary electron is produced. – During SE production, an inner shell electron is emitted from the atom leaving a vacancy. – Higher energy electrons from the same atom can fall into the lower energy hole. This creates an energy surplus in the atom which is corrected by emission of an outer shell (low energy) electron • Use – AE have characteristic energies that are unique to each element from which they are emitted. – Collect and sort AE according to energy to determine composition. – AE have very low energy and are emitted from near surface regions. X-rays • Formation – Same as AE. Difference is that the electron that fills the inner shell emits energy to balance the total energy of the atom. • Use – X-rays will have characteristic energies that are unique to the element(s) from which it originated. – Collect and sort signals according to energy to yield compositional information. – Energy Dispersive X-ray Spectroscopy (EDS) Foundation of XPS (X-ray photoelectron spectroscopy). XPS can be used to determine the “state” of an atom and to identify chemical compounds. Transmitted electrons • Can be used to determine: – thickness – crystallographic orientation – atomic arrangements – phases present – etc. • Foundation for Transmission Electron Microscopy (TEM) Specimen Interaction Volume • Volume where electron beam - specimen interactions occur. • Depends upon: – Z of material being examined • higher Z materials absorb more electrons and have smaller interaction volume – Accelerating Voltage • higher voltages penetrate further into the specimen and generate larger interaction volumes – Angle of incidence of electron beam • larger angle leads to a smaller interaction volume
  • 9. ~1 nm – Auger electrons ~5–50 nm Secondary electrons ~300 nm Backscattered electrons ~50 nm TEM specimen thickness ~1.5 µm X-rays ??? X-ray resolution Z = 29 (Cu), Accelerating voltage = 20kV INCIDENT BEAM 10 Å - 30 Å – Auger electrons ~100 Å Secondary electrons <1µm - 2µm Backscattered electrons ~5 µm X-rays ~1.0 µm X-ray resolution Z = 28 (Ni), Accelerating voltage = 20kV INCIDENT BEAM ~50 nm TEM specimen thickness • Interaction volume is larger for materials that have lower atomic numbers and for higher incident beam energies! ~50 nm Increasing atomic number (Z) Increasing incident energy (E0) Z1 Z2 Z3 Z1 < Z2 < Z3 “Instruments of the trade” • Primary Instruments – Transmission Electron Microscope (TEM) – Scanning Electron Microscope (SEM) • Variants – Electron Probe Microanalyzer (EPMA) – Scanning Transmission Electron Microscope (STEM) – Environmental SEM (ESEM) – High Resolution TEM (HRTEM) – High Voltage TEM (HVTEM) – etc...
  • 10. TEM • Patterned after transmission light optical microscopes • Yield Following Information: – Morphology • Size shape and arrangement of particles, precipitates, etc. – Crystallographic information • Atomic arrangement – Compositional Information • If proper detector is present The TEM is analogous to the transmission LOM Source Condenser system Stage Objective lens Final imaging system Observation and recording system x y z + Tilt & rotation Vacuum column Apertures x y z 1 in 1020 electrons are collected <1 in 106 photons are collected TEM LOM Phosphor screen; film; digital recording system Eyepiece or TV screen; film; digital recording system Light source Electron gun Components of the TEM • Source – filament plus anode plates with applied accelerating voltage. • Condenser Lenses – electromagnetic lenses adjusted by lens currents not position. • Specimen Stage – allows translations and tilts. • Objective Lens – usually <50X. • Imaging System – multiple electromagnetic lenses below the objective: set magnification, focal plane (image vs. diffraction pattern). • Observation – fluorescent screen, plate film, CCD camera. x y z + Tilt & rotation Vacuum column 1 in 1020 electrons are collected Phosphor screen; film; digital recording system Electron gun Adapted from B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) Condensor Lenses ELECTROMAGNETIC lenses focus the electron beam to as small a spot as is possible. They are equivalent to convex lenses in optical lens systems. Light Source OPTICAL LENS Cu coils Soft Fe pole pieces N S N S Electron Source MAGNETIC LENS Adapted from B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
  • 11. Electron Lenses • Use electrostatic or electromagnetic fields to focus beams of charged electrons. • ELECTROMAGNETIC – Most lenses are of this type. Consist of Cu wire coils around soft Fe cores. Sometimes an Fe pole-piece is used to “shape” the field. • ELECTROSTATIC – Unusual. Only common example is the Wehnelt aperture in the electron gun. Electrons in Magnetic Fields A. Electrons moving through a perpendicular magnetic field experience perpendicular forces. B. Electrons moving parallel to a magnetic field are unaffected. C. Electrons moving nearly parallel to a magnetic field take a helical path around the magnetic field direction (i.e., the microscope axis). electron Magnetic field Force on electron is out of plane No Force Electrons follow a helical path unaffected Nearly aligned electron A B C Adapted from B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) Trajectories in Electromagnetic Lenses • When you adjust the magnification (and the focal length), you modify the lens strength by adjusting the current in the electromagnetic lens coils. • Since the magnetic field is changed, so are the helical trajectories. • This ultimately leads to image rotation which must be corrected for or calibrated. Rotated image Object Cu coils Soft Fe pole pieces N S N S Image is inverted and rotated Electron Source MAGNETIC LENS Beam Tilting and Translation • The electron beam can be positioned for fine measurements (spot modes) or scanning (SEM, STEM) BEAM TRANSLATION BEAM TILTING APPARENT ORIGIN APPARENT ORIGIN Upper scan coil Lower scan coil
  • 12. Condenser Alignment • In TEM it is important to center the condenser aperture about the optical axis. • If the aperture is off center, the beam is displaced away from the axis when the condenser lens is focused. • You can try condenser alignment at: http://www.matter.org.uk/tem/condenser_alignment.htm Condenser Astigmatism • Astigmatism in the condenser lenses distorts the beam to an elliptical shape either side of focus and prevents the beam being fully focused. It is corrected by applying two orthogonal correction fields in the x and y directions. • You can try stigmator adjustment at: http://www.matter.org.uk/tem/condenser_astigmatism.htm Objective Aperture • Depending upon where you position the objective aperture and whether or not you tilt the beam, a bright field image, a dark field (i.e., scattered) image, or a diffraction pattern can be collected. • You can look at this in more detail at: http://www.matter.org.uk/tem/dark_field.htm B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) AND http://www.matter.org.uk/tem/stem_images.htm
  • 13. B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
  • 14. B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) BF TEM Micrograph and SADP Grain structure Crystallographic information [111] zone axis NiAl-1.0Hf
  • 15. 250 nm 1hr NiAl-1.0Hf BF TEM Micrographs Precipitate distribution Defect structures 4hr NiAl-1.0Hf 100 nm β′ Defect Structures Dislocations imaged in NiAl-0.5Zr single crystals deformed at elevated temperatures. Nano-scale X-ray Analysis β′ -NiAl EDX spectrum β′ -precipitate β′ HAADF (Z-contrast imaging) analysis 2 1 Ni2AlHf Hf rich region • Hf segregates to GB’s in as-deposited & annealed coatings. • Hf precipitates as β´-Ni2AlHf phase after annealing. NiAl-1.0Hf NiAl-1.0Hf As-deposited Annealed 1000°C/1h 20 nm 200 nm
  • 16. How does a TEM work? • Pass a focussed beam of electrons through a thin foil • As beam passes through sample, it is scattered • Project the transmitted (scattered) beam onto a phosphor screen to form an enlarged image • Modes: – Bright Field/Dark Field modes for visualization of structure and defects – Selected Area Diffraction/Convergent Beam Diffraction for crystallographic information Electron Beam Thin Foil < 500 nm thick Electron Gun Anode Plate First Condenser Lens (C1) Second Condenser Lens (C2) Condenser Aperture Sample Objective Lens Objective Aperture Selected Area Aperture First Intermediate Lens Second Intermediate Lens Projector Lens Phosphor Screen Used to magnify image. Schematic Representation of optics in a TEM Technical Details • Produce a stream of monochromatic electrons in the electron gun • Focus the stream into a small coherent beam using C1 and C2 – C1 determines the “spot size” (i.e., size of electron probe) – C2 changes intensity or brightness • Use condenser aperture to restrict the beam • Part of the beam is transmitted through the sample • Focus transmitted portion using the objective lens to form an image • Objective and selected area apertures are used to restrict the beam further – allows examination of diffraction from specific atoms, crystals, features… – SAD, CBD • Enlarge image with intermediate and projector lenses
  • 17. SEM • Patterned after reflecting light optical microscopes • Yield Following Information: – Topography • Surface features of an object. Detectable features limited to a few nanometers depth. – Morphology • Size shape and arrangement of particles, precipitates, etc – Compositional Information • Elements and compounds the sample is composed of – Crystallographic information • Possible using new techniques (OIM/BKD) x y z Vacuum column x y z TEM SEM Electron gun Condenser system Stage Objective lens Final imaging system Observation and recording system Probe lens Stage Scan coils Time base Signal amplifier CRT Condenser lens Also tilts and rotates Also tilts and rotates In the SEM you use secondary signals to acquire images. Electron gun Detector and processing system Components of an SEM • Source: – same as TEM but lower V • Condenser: – same as TEM • Scan Coils: – raster the probe • Probe Lens: – lens that forms a spot at the specimen surface • Detector & Processing System: – collects signals such as X-rays and electrons as a function of time and position. – Provides digital images for real-time viewing, processing, and storage. x y z Probe lens Stage Scan coils Time base Signal amplifier CRT Condenser lens Also tilts and rotates Electron gun Detector and processing system Schematic Representation of the optics in an SEM Electron Gun Anode Plate First Condenser Lens (C1) Second Condenser Lens (C2) Condenser Aperture Sample Objective (probe) Lens Objective Aperture Scan Coils Detector
  • 18. SEM: Technical Details • Produce a stream of monochromatic electrons in the electron gun • Focus the stream using the first condenser lens – Coarse probe current knob • The beam is constricted by the condenser aperture (eliminates high-angle electrons) • Second condenser lens is used to form electrons into a thin, tight, coherent beam. – Use fine probe current knob • Use objective aperture to limit beam (i.e., eliminate high-angle electrons) • Scan coils raster the beam across the sample, dwelling on the points for a predetermined period of time (selected using scan speed) • Final objective lens focuses beam on desired region. • When beam strikes the sample, interactions occur. We detect what comes out of the sample. Secondary electron mode Could get additional information in backscattered or mixed mode 10 µm Substrate IDZ Coating γ′ γ γ+γ′ γ+γ′ 40 50 60 70 80 0 10 20 30 40 0 10 20 30 40 -15 -10 -5 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 γ' Substrate γ + γ' Composition (at%) Ni-AD Ni-Annealed Ni Al-AD Al-Annealed Al Pt-AD Pt-Annealed Pt Hf-AD Hf-Annealed Hf Distance (µm) Figure 4.8. Composition profiles of the major elements measured by WDS in an EPMA.
  • 19. X-ray Analysis in the SEM EDS or WDS Elemental diffusion profile NiAlPtHf coatings on CMSX-4 substrates for a temperature of 1150°C and times from (a) 20 hrs and (b) 100 hrs. Al O Ni Pt Hf Al Ni O Pt Hf Al2O3 + ??? ??? • Change magnification by changing the scan area • Focus by changing the objective lens current • The probe current/spot size are controlled by the condenser lens current Useful things to remember about SEM operation B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
  • 20. B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) Microanalysis in Electron Microscopy • Characteristic X-rays are always generated by interactions between the incident electron beam and the sample. They constitute a fingerprint of the local chemistry. • Collect X-ray signal to determine local chemistry • Common Techniques: – Wavelength-Dispersive Spectrometry/Spectroscopy (WDS) – Energy-Dispersive Spectrometry/Spectroscopy (EDS) WDS • Yields best discrimination of emitted X-ray signal • Use a series of bent crystals to cover the range of wavelengths of interest • Scan wavelengths within each range by rotating the crystal and moving the detector while keeping the position of the crystal fixed. • X-rays are collected from the sample at a fixed angle. The angle at the collecting crystal will vary with 2θ and the diameter of the focusing circle will change
  • 21. Specimen X-ray Detector Receiving slits Bent crystal Radius = 2R R Focusing Circle WDS WDS continued • All peaks are scanned sequentially. You cannot record more than one characteristic line at once unless there are multiple spectrometers available. • EPMA! • Data is collected sequentially. Takes more time but the results are much more precise than EDS. Resolution of M lines in a WDS spectrum of a superalloy (Oxford Instruments) EDS vs. WDS • Pulse height is recorded by the detector. It is related to the energy of the photon responsible for the pulse. • Solid-state detectors are generally used. • EDS is faster than WDS • Problems with EDS: – Poor discrimination or energy resolution. WDS systems are much better, particularly when characteristic lines from different elements overlap. – Need a windowless or thin window detector to pick up light elements. – WDS is more quantitative
  • 22. The BSE sampling volume is large which limits resolution Multiple elements of BSE detector “Overlapping shadows” Reduced resolution SE image Single segment of BSE detector Can use BSE signal in conjunction with SE signal to yield enhanced topographical information Images • Topography and atomic number influence signal. • Backscatter coefficient: – η = -0.254 + 0.016Z - 1.86x10-4Z2 + 8.3x10-7Z3 • Atomic number contrast: – C = (η1 - η2)/ η1 TOPOGRAPHIC CONTRAST TOPOGRAPHIC CONTRAST Signals from 2 detectors B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
  • 23. B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
  • 24. B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)
  • 25. B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005) B.D. Huey, MMAT322 Lecture Notes, University of Connecticut (2005)