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Nano-technology and
Department of Electrical and
Computer Engineering,
University of Tehran
Measurement in Nano
Electron microscopes, diffraction,
Transmission electron microscopes,
Scanning electron microscopes,
Tunneling microscopes,
Scanning positioning microscopes,
Atomic force microscopy,
Optical microscopes,
Dark field phase microscopes,
Depth of focus, poor for electron microscopes,
Small apertures, loss of electron beam
First TEM
• Ruska and Knoll, 1930!
Fundamental of TEM
Diffraction of electrons
• Matter-wave nature
of electrons,
• λ=h/p where
– h: Planck’s constant
– p: momentum, (mv)
– λ:wavelength of
• E=1/2 (m0v2) or
– λ= h/(2m0eV)0.5
• or λ=
Relativistic effects
• V: accelerating voltage, non-relativistic
– 100kV 0.0038nm,
– 200kV 0.0035nm
– 400kV 0.0023nm
• Relativistic wavelength
– 200kV 0.0033nm, 2*108m/s
– 400kV 0.0016, 2.5*108m/s
– Increase in mass m/m0= 1.78
• High speeds, close to speed of light
• Microscopes with ultra-high resolution,
Various effects
Electron scattering
• Interaction of electron with a
single isolated atom,
• Scattering angle (θ) small
• Solid angle, , steradian(sr)
• σ=πr2 , cross-section,
• d=2πsinθ dθ
• dσ/d = 1/(2πsinθ) dσ/dθ
– Differential cross section
• σ=πrelast2 , relast = Ze/Vθ
• σt= σelas + σ inelas
Diffraction of light
The scattered waves are in-phase when the path difference is a nλ
L=d sinθ, d: spacing of slits
Detector is placed far away at angle of θ
Two wavelets traveling in direction (r) are out of phase by 2πL/λ
This difference is called a “phasor”
Diffraction patterns
• Five-slit aperture,
• If phasors are
360/5 (o) apart,
the resultant
vector is zeromagnitude.
• Second zero
happens at 144o,
Finite width slit
Airy diffraction
• Visible light
produced by
0.5mm diameter
• Airy rings:
resulted from
diffraction from
small aperture.
Angles in TEM
• Coulomb scattering
from atoms and
• Higher energy
electrons, less
• r=Ze/Vθ
• Smaller distances,
more scattering
• Higher energies, less
Wave scattering
• Two waves traveling,
incident and scattered.
• Incident wave,
• Incident wave could be set
at Z-axis
• Reflected (scattered) wave:
Ψsc(r)= Ψ0 f(θ)/r exp(ikr)
• Summation of both waves
must be valid in SE.
• Ψt(r)= Ψ0{exp(iKIr) +if(θ)/r
Atomic factor
• |f(θ)|2= dσ(θ)/d
• dσ(θ)/d =e4Z2/(16E02sin4θ/2)
• Where E0 in eV, is the incident
energy of electrons
• dσ(θ)/d
λ4Z2/(64π4a02(sin2θ/2 + θ02/2)2)
• a0=h2ε/(πm0e2), Bohr radius,
around 0.5Ǻ
• θ0 describes the electron-electron
scattering, about 2 degrees,
• When θ bigger than θ0 nuclear
scattering is dominant.
• Diffraction of waves in terms of reflection of a
plane wave at an angle of “theta”.
• The path difference is AB+BC,
• Under Brag condition this path difference is a
multiple of wavelength.
Brag diffraction
• K in this image is
the same as “g” in
other notations.
Real image, diffraction
Diffraction patterns
• Single crystals, regular patterns,
• Poly-crystals, dotted pattern
• Many-crystals, rings
TEM in Electronics
• Top: TEM image of 500nm,
silicon epitaxy, the bending
lines are evident
• Right: image of small
Ewald sphere
• Reciprocal Lattice,
Condition: Exp(iK.R)=1
• K.R=2nπ, R: translation
vector, K defines the RL.
• Incident beam, k=1/λ
• Smaller λ, larger radius,
• Brag conditions, ki-kd=g
• “g” a vector in reciprocal
Finite specimen
• Extinction error (s) or deviation parameter.
• Diffraction occurs even without Brag’s condition
• When ki-kd=g+s, the intensity of the diffraction
spots depends on the value of “s”.
Finite thickness
• Kinetic theory, specimen thickness is split into slices of atomic
• Diffraction from solid is the summation of all slices.
• Similar to diffraction from slit with a limited width,
• Ig(s)=(π/ξg)2 (sin(πts)/πs)2 where ξg is the extinction distance.
– Ig is the intensity of the diffracted beam.
• Thinner specimens, more deviation from an ideal crystal or Brag
• More chance of electron penetration through the specimen to
measure the diffracted beam
Diffraction patterns
• Diffraction patterns of silicon along 110 direction.
• By increasing the sample thickness, DP becomes hazier.
Convergent beam
Image formation
Energy losses
• Auger electron,
secondary electron
• Emission from
• Characteristics of
the material.
• High adsorption
• Surface effect
Inelastic scatterings
• Phonon, lattice
• High Z atomic systems
• Mean free path,
• Hamper diffraction
• Cooling the specimen
for better imaging.
• Plasmon, longitudinal
electron wave
• Resulted from impact
of high energy
• Similar to acoustic
• Electron gas in highly
conductive metals,
• Mean free path about
• Incident electron leads
to a promotion of
electrons from V.B to
• The return on this
electron leads to a bandto-band recombination.
• For a direct gap
semiconductor, a
radiative recombination
is observed.
• Photons with the value
of the B.G. are emitted.
Electron guns in TEM
• Tungsten hair-pin
tip: easy to use,
• Low vacuum
conditions, high
• Thermionic
• Low current
Crystalline sources
• LaB6 crystal
• Undersaturated
emission, mostly
from corners,
• Saturated, a
uniform and
coherent emission.
Field-Emission guns
Coherent and high current density
Two anodes to extract and converge the beam.
Need for ultra-high vacuum technologies,
Applications in SEM.
Various definitions
• Depth of focus:
depth of sharpness
in the image plane,
• Depth of field: depth
of sharpness in the
object space
• αim=dim/Dim
• βob=dob/Dob
• It can be shown
Dim=dob/ βob MT2
TEM images of nanoparticles
• TEM of nickel
particles on silicon
• (a) Bright field
image, (b) Diff.
• (c) aperture
filtered, (d)
improved aperture
• (e) processed
image, (f) oxide