Download Magnetic Lenses, Interactions of Electrons with Matter

Magnetic Lenses
Magnetic fields can displace electrons
Magnetic field can be produced by passing an electrical current through coils of wire
Magnetic field strength can be increased by using a soft ferromagnetic core like Fe.
F = e (v x B)
Current in coil ~ 0 – 1 A
Electron initially unaffected by Bax, but feels
force Bradev – pushes it into a helix.
Now v has component perpendicular to axis –
Bax pushes it into tighter helix.
Image rotates!
Iron core/shroud
Magnetic field nearly
parallel to electron
trajectory
It is impossible to produce a perfect EM lens – only approximates to
Bax grows as electron approaches middle of gap –
path narrows and focuses about optic axis
Maxwell lens when electrons are close to axis.
MSE 321 Structural Characterization
Electrons
Interactions with Matter
N = number of scattering particles per unit volume
Scattering cross section, σ: area which the scattering particle appears to
present to the electron
p = N σdx
p = probability of scattering
N = number of particles per unit volume
dx = distance through specimen
Mean free path, λ: average distance an electron will travel before
being scattered in a particular way
λ = 1/(N σ)
p (n) = (1/n! )(x/λ)nexp(-x/λ)
Poisson equation - Assumes multiple scattering by same process – not very accurate
Interaction volume: region into which the electrons penetrate the specimen
MSE 321 Structural Characterization
Electrons
Interaction Volume
d
Incident (primary) beam
SEM
TEM
Secondary electrons
Backscattered electrons
≈1µm
≈100nm
(d 2+b 2)½
X-rays
Numerous scattering events widen diameter of interaction volume
Depth increases as Voltage increases
Depth decreases as Z increases
b = 0.00807(Z /E )(Nv )1/2t 3/2 (TEM)
b [nm], Z [atomic number], E [keV], Nv [atoms/nm3], t [nm]
MSE 321 Structural Characterization
Electron Scattering
Inelastic Scattering: Any process which causes the primary electron to lose a detectable amount of energy
Elastic Scattering:
Any scattering process which results in no measurable change to the energy of the
primary electron
Backscattered
electron
primary
electrons
cathodoluminescence
Characteristic
x-ray
or
Auger
electron
ejected secondary electron
primary
electron
MSE 321 Structural Characterization
ejected secondary electron
primary electron
Inelastic Scattering
Phonon scattering
~ 1 eV per interaction, but heating can be considerable, especially at high kV.
Not a problem in good conductors (∆T ~ 10°C), but can melt Al2O3 (Tm > 2000°C)
Large deflection (high-angle) of electron (~10°) – diffuse intensity.
Phonon: quantum of atomic vibration
Plasmon scattering
~ 5 – 30 eV per interaction and small mean free path (very common)
Contributes some diffuse intensity around transmitted spot.
Important in Auger and EELS studies.
Plasmon: wave in the sea of
conduction-band electrons
Single valence-electron excitation
~ 1 eV per interaction, but very large mean free path (microns) and small scattering angle
Primary electron transfers energy to a single conduction-band electron rather than entire sea
Not exploited in electron microscopy
Inner shell excitation leading to characteristic x-ray production
100s – 1000s eV per interaction, large mean free path (µm)
Primary electron transfers enough energy to an inner electron (K, L) to knock it out
Outer electron drops down to fill the hole releasing a characteristic quantum of energy
λ increases as V increases (more likely to pass through without interacting)
λ increases as Z increases (critical energy, Ec, required to produce an x-ray increases)
283 eV for Carbon K (Z = 6), 69,508 eV for Tungsten K (Z = 74)
Produced throughout interaction volume, and virtually all escape from the surface
Smallest region which can be analysed by SEM ~ 1 µm.
Fluorescence yield , w = Z 4 / (Z 4 + c)
Much higher for large Z
σα
1
EcEo
c ~ 106 for K
MSE 321 Structural Characterization
Inelastic Scattering
Cont…
Inner shell excitation leading to characteristic (Auger) electron production
Alternative to x-ray emission
Primary electron loses some energy to inner-shell electron, which is knocked out of the atom
One outer electron falls into the hole left behind
Another outer electron carries off ∆E as kinetic energy
Yield increases for low Z, so have low energies and only escape from top ~1 nm of specimen
Important surface analysis technique, but require very high vacuum sytem
Auger electrons are outer electrons and so contain information about bonding
Auger yield = 1 - w
Outer shell excitation leading to cathodoluminescence (CL)
Similar to inner-shell excitation, but energy of resulting photon is lower (typically visible or UV)
Emission from semiconductors and insulators is modified by presence of defects (e.g., dislocations)
Excitation of outer electrons leading to emission of low-energy secondary electrons
Energies < 50 eV, so can escape from top ~ 10 nm of specimen - topographical
either primary electrons multiply scattered or (more probably) electrons created by a process above
Edges, corners, and areas tilted towards detector appear bright
yield can be > 1
primary beam
high yield
shadowing
low yield
specimen
MSE 321 Structural Characterization
Inelastic Scattering
Bremsstrahlung x-rays (bremsen = to brake, Strahlung = radiation)
primary electrons are decelerated and deflected by Coloumb field of atoms in specimen – KE transformed into x-rays
Mixture of x-rays with many wavelengths – no use in microanalysis – unwanted background radiation
MSE 321 Structural Characterization
Inelastic Scattering
Absorption
Penetration depth/mean free path determines depth of specimen sampled
Varies with keV and material, but typically shorter than for x-rays
When specimen is very thick, you won’t see an image.
Have electrons been absorbed??
π
θ
Extinction distance: ξ g = Vc cos
λFg
seff = s 2 +1 ξ2g
ξeff
=
g
ξ
2

(s = 0)
g
1+ ξg2s2
π  sin2(πts)
ξ  (πs)2
 g
Kinematic: I g = 

Dynamical: I g =  π
Define imaginary component of extinction distance:


2
ξ 
 g
sin2(πtseff)
(πseff)2
1
1
1
=
+i ′
ξgeff ξg
ξg
2

 sin2(πts )
eff
Dynamical + Absorption: I g =  πeff
 ξg 
(πseff)2
Usable thickness limited to ~5ξg
MSE 321 Structural Characterization
for s = 0,
increases as
t
∞
sinusoidal
variation t
ξg′ ~ 10ξg
Fudge factor!
∞
Inelastic Scattering
Absorption
Penetration depth/mean free path determines depth of specimen sampled
Varies with keV and material, but typically shorter than for x-rays
When specimen is very thick, you won’t see an image.
Have electrons been absorbed??
ξ0
= Linear absorption (overall decrease in intensity with increased t)
ξ0′
ξg
= Anomalous absorption (selective absorption of certain electron waves)
ξg′
Kinematical (|s| >> 0): fringes are closely spaced and limited to thin regions near the hole.
Fringes are stronger in the DF than in BF (i.e., they are non-complimentary), and the contrast
from defects is low.
Dynamical (s = 0): fringes are broader with reasonably complimentary BF and DF images.
Defect contrast is strong and the best defect images occur just as fringes damp out.
MSE 321 Structural Characterization

2
π  sin2(πtseff)

ξ
(πseff)2
 g
I g = 
ξg = πVc cos θ
λFg
seff = s 2 +1 ξ2g
(s = 0)
Intensity of Diffracted Beam
Rocking Curve
t/ξg = 1.5
ξg/ξg' = 0.1
ξg/ξo' = 0.1
-0.04
-0.02
0
0.02
0.04
Deviation Parameter (nm-1)
Ig is periodic in t and seff.
If t is constant and seff is varied
MSE 321 Structural Characterization
bend contours
Dark-Field bend
contours in
Pb3Nb2O8
Thickness Fringes
 π
I g = 
ξ
 g




2
sin2(πtseff)
(πseff)2
seff = s2 +1 ξ2g
Intensity of Diffracted Beam
If seff is constant and t is varied
thickness fringes
ξg
ξg = 100 nm
ξg/ξg' = 0.1
ξg/ξo' = 0.1
w = 0 (s = 0)
Effective extinction distance
ξw
=
g
0
100
200
300
Thickness (nm)
ξ
g
1+ w 2
w = sξg
400
When other diffracted
beams are present, the
effective ξg is reduced.
ξg
MSE 321 Structural Characterization
Elastic Scattering
Rutherford Scattering
Coulombic interactions between primary electrons and atoms in specimen
Probability (per unit area) of scatter:
n = atoms per unit volume in target
L = thickness of target
Z = atomic number of target
e = electron charge
k = Coulomb’s constant, 1/(4πεo)
r = target-to-detector distance
E = kinetic energy
θ = scattering angle
Strongly forward-peaked
(small-angle scatter much more probable than large-angle scatter)
MSE 321 Structural Characterization
Electron Microscopes
All have electron gun (source), condenser lens system, and signal detection
SEM also has scanning coils
Add EDS and possibly WDS
it becomes an analytical SEM
Hitachi 4500 SEM
Add several WDS spectrometrs
and it becomes a Microprobe (EPMA)
TEM also requires objective and projector lenses
Add scan coils and it becomes
a scanning TEM (STEM)
JEOL 2100 HR TEM
MSE 321 Structural Characterization
Add EDS/EELS and it becomes
an analytical TEM/STEM