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Diffraction of “low energy” electrons from free-standing transmission gratings
Ben McMorran and Alex CroninUniversity of Arizona
200 nm
free-standing silicon nitride gratings
note the cross section
SEM basics
objective lens
sample
~3mm to 42mm
100 µm aperture
SEI
+10V
detector
PMT
phosphor
~ +500V
SEM basics
sample
SEI
+10V
detector
PMT
phosphor
~ +500V
objective lens
~3mm to 40mm
100 µm aperture
basic setup to observe diffraction
objective lens
diffraction grating
4 µm diameter tungsten wire
~30 mm
SEI
~40 mm
100 µm aperture
images of 4 micron wire through diffraction grating
4 keV beam
twist = -10±2°
1.5 keV beam
twist = 5±3°
images of 4 micron wire through diffraction grating
4 keV beam
twist = -10±2°
1.5 keV beam
twist = 5±3°
images of 4 micron wire through diffraction grating
1.5 keV beam
twist = 5±3°
4 keV beam
twist = -10±2°
images of 4 micron wire through diffraction grating
• spacing between orders Δs E-1/2
• why is there asymmetry?
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0
-40 -20 0 20 40
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0
-60 -40 -20 0 20 40 60
4 keV beam
twist = -10±2°
1.5 keV beam
twist = 5±3°
grating geometry
w
l
gold coating
grating bar
grating k vector
grating geometry
(,z)
z
grating geometry
z
(,z)
grating geometry
(,z)
z
grating geometry
(,z)z
grating geometry
(,z) z
grating geometry
(,z)z
grating geometry
(,z)
z
grating geometry
(,z)
z
calculation of phase due to image charge
21
11
2),,,(
rr
qezU o
o{ z
(,z)
r1
r2
calculation of phase due to image charge
21
11
2),,,(
rr
qezU o
path
Ldt1
o{ z
(,z)
r1
r2
calculation of phase due to image charge
2/
2/
),;,(1
,;l
l
ooimage dzzUv
21
11
2),,,(
rr
qezU o
path
Ldt1
o{ z
(,z)
r1
r2
calculation of phase due to image charge
2/
2/
),;,(1
,;l
l
ooimage dzzUv
21
11
2),,,(
rr
qezU o
path
Ldt1
o{ z
(,z)
r1
r2
2/
2/
w
w
oi
n de nimage
no
n
2where
calculation of phase due to image charge
2/
2/
),;,(1
,;l
l
ooimage dzzUv
21
11
2),,,(
rr
qezU o
path
Ldt1
o{ z
(,z)
r1
r2
2/
2/
w
w
oi
n de nimage
no
n
2where
,,,,,* lwEII nnnn
calculation of phase due to image charge
2/
2/
),;,(1
,;l
l
ooimage dzzUv
21
11
2),,,(
rr
qezU o
path
Ldt1
o{ z
(,z)
r1
r2
2/
2/
w
w
oi
n de nimage
no
n
2where
,,,,,* lwEII nnnn
?)(E E = 1 keV v ~ 107 m/s τ ~ 10-14 sec
target
twist axis
Electron beam
So, if we measure diffraction at different twist angles…
…we ought to see something like this:
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0
x106
-40 -20 0 20 40
0 order
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20
10
0
x106
-40 -20 0 20 40
+1 order-1 order
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10
5
0
x106
-40 -20 0 20 40
+2 order-2 order
-0.10
-0.05
0.00
0.05
0.10
-40 -20 0 20 40
(I1 - I-1)/(I1 + I-1)
-0.2
-0.1
0.0
0.1
0.2
-40 -20 0 20 40
(I2 - I-2)/(I2 + I-2)
500 eV
grating
1st aperture
target (4 µm wire)
sliding platform
3rd aperture
description of apparatus
description of apparatus
grounding strap
twist lever
tilt stage
diffraction profiles - comparison
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-60 -40 -20 0 20 40 60
twist = 5±3°, 500 eV
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-60 -40 -20 0 20 40 60
twist = 10±2°, 500 eV
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-60 -40 -20 0 20 40 60
twist = -10±2°, 500 eV
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-60 -40 -20 0 20 40 60
twist = 0±2°, 500 eV
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0
-60 -40 -20 0 20 40 60
twist = 5±3°, 1.5 keV
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-60 -40 -20 0 20 40 60
twist = 10±2°, 1.5 keV
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-60 -40 -20 0 20 40 60
twist = -10±2°, 1.5 keV
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-60 -40 -20 0 20 40 60
twist = 0±2°, 1.5 keV
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-40 -20 0 20 40
twist = 10±2°, 4 keV
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0
-40 -20 0 20 40
twist = 5±3°, 4 keV
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0
-40 -20 0 20 40
twist = -10±2°, 4 keV
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80
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0
-40 -20 0 20 40
twist = 0±2°, 4 keV
10±2°:
5±2°:
0±2°:
-10±2°:
500 eV 1.5 keV 4 keV
data refining
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0
-40 -20 0 20 40
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150
100
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0
-60 -40 -20 0 20 40 60
4 keV beam
twist = -10±2°
1.5 keV beam
twist = 5±3°
4 keV beam
twist = -10±2°
160
140
120
100
80
60
40
20
0
-40 -20 0 20 40
200
150
100
50
0
-60 -40 -20 0 20 40 60
1.5 keV beam
twist = 5±3°
data refining
1.5 keV beam
twist = 5±3°
• boil images down to In data – compare to theory
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0
-40 -20 0 20 40
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150
100
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0
-60 -40 -20 0 20 40 60
4 keV beam
twist = -10±2°
data refining
conclusion
• have seen diffraction of electrons with energies down to 500 eV through a transmission grating
• have seen asymmetry in diffraction pattern due to interaction with grating
• a simple model using image charges seems to explain asymmetry
goals
• more angles with better precision
• more energies
• include detector capable of measuring absolute intensity of diffraction orders (not just relative intensity)
• search for energy-dependent permittivity
