1
Free-standing, metal-coated transmission gratings for
electrons
Glen Gronniger1, Brett Barwick1, Tim Savas2, Alex Cronin3, Dave Pritchard2, and
Herman Batelaan1
1
Department of Physics and Astronomy, University of Nebraska—Lincoln, 116 Brace
Laboratory, PO Box 880111, Lincoln, Nebraska 68588-0111, USA
2 Research
Laboratory of Electronics, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Room 36-413 Cambridge, Massachusetts 02139-4307, USA
3
Department of Physics, University of Arizona, 1118 E 4th St. Tucson, Arizona 85721,
USA
PACS numbers: 0.3.75.-b, 41.85.Ct, 34.80.Pa
Electron diffraction from free-standing nano-fabricated transmission gratings is
demonstrated, with low energies ranging from 125 eV to 25 keV. The effect of
image charges in the grating bars is detected by analyzing the diffraction patterns.
For 125 eV electrons, the grating broadens each diffracted beam. Although the
mechanism is unknown, the broadening is consistent with a model calculation
based on path integrals that includes a random potential with a characteristic
length scale of 250 nm.
Pioneering work in atom optics [1] led to the development of 100-nm-period silicon
nitride transmission gratings with free-standing bars [2]. Such gratings have since been
used for Na atom interferometry [3], He2 molecule diffraction [4] and Bucky ball
diffraction [5]. In this Letter we report that free-standing gratings can also diffract lowenergy electron beams. This technology permits electrons with kinetic energy as low as
125 eV to be diffracted and transmitted with high efficiency as we have demonstrated
2
experimentally. Interactions between electron de Broglie waves and the grating
structure do affect the diffraction patterns, but our analysis suggests that Mach-Zehnder
[3] and Talbot-vonLau [6] interferometry is now possible with low energy electrons.
This opens new possibilities for electron physics as indicated in the conclusion.
Until now, transmission diffraction of low-energy electrons has been problematic. For
example, solid crystals absorb low energy electrons. Thin collodion films coated with
gold stripes were developed [7], but these also suffer from low transmission. In
collodion, 125 eV electrons undergo inelastic scattering in a distance much less than 10
nm [8]. Free standing transmission gratings avoid this problem because the physical
channels between the bars transmit low energy electrons without inelastic scattering.
Free standing gratings were pioneered by Jönsson [9], but their regularity was poor.
Now, high transmission combined with good quality diffraction of low-energy electrons
has been achieved for the first time by using metallized silicon nitride nanostructures
with 50-nm channels between free standing bars.
We have diffracted electrons with kinetic energy ranging from 125 eV to 25 keV which
corresponds to de Broglie wavelengths ranging from 110 pm to 7.8 pm. Electron beams
for this study have been generated by two methods: first in a dedicated beam apparatus
described below, and secondly with the converging beam of an electron microscope
described later. In each case the total transmission is about 30%, which is certainly high
enough to envision future interferometer experiments using multiple gratings. A major
challenge for this plan is to understand the effect of image charge potentials and any
random potentials that may cause decoherence. An analysis including these potentials
is discussed after the data presented below.
Our diffraction experiment is straightforward: electrons were emitted from a
thermionic source at energies ranging from 125 eV to 1000 eV, with an energy width of
approximately 1eV. This electron beam was collimated by a 5 m wide molybdenum
3
slit followed 25 cm downstream by a 2 m wide by 10 m tall slit that was made using
a focused ion beam to mill a 100 nm thick silicon nitride substrate. The second slit was
coated first with titanium and then with gold on both sides. The angular spread of
electron trajectories that can travel straight through the collimation slits is
(5  2)  m
 geom 
 2.8  105 rad . This agrees with the observed angular spread for our
25cm
500 eV electron beam which produces a 7 m wide spot (FWHM) in the plane of
detection, located 31 cm downstream of the 2  m slit. The geometrical transverse
momentum spread through the second slit is pgeom ( E )   geom 2mE . The
corresponding transverse coherence length is lt 
0.89h
 2 m [10], which means
p(500eV )
that the second slit is coherently illuminated. The roughly 125 nm thick silicon nitride
grating (Fig. 1) was placed 7 cm downstream from the second collimation slit. We infer
from this geometry that the electron spot size on the grating is approximately 4  m , and
coherent over (at least) 2  m . This is consistent with the transverse coherence length
inferred from the observed resolution, R, of the diffraction peaks. The resolution is
defined as the ratio of the peak separation over the peak width and d is the grating
135
100nm  1 m (at 500 eV,
periodicity. The coherence length is given by lt  Rd 
11
see figure 2). This means that the electron beam is nearly diffraction limited, and the
grating does not affect the coherence length much at higher energies. Beam broadening
 diff

E
 n dB  n
 n103 , where n
due to the energy width of the electron beam,
 diff
dB
2E
is the diffraction order, is negligible (the associated longitudinal coherence length and
time are about 50 nm and 5  10 15 s ). The faces of the grating had been metallized with
a 2 nm layer consisting of 60% gold and 40% palladium, while the grating channel
walls had been covered with a layer approximately 0.3 nm thick. The detector is a
channeltron placed behind another 5  m molybdenum slit. By controlling the
thermionic source current, we limit the count rates to 106 electrons per second to protect
the channeltron. Combined with our electron velocity of 107 m/s this means that only
4
one electron is in our system at any one time, which excludes the possibility of any
electron-electron interaction. The apparatus is operated in a 10-7 Torr turbo pumped
vacuum system and is shielded from the earth’s magnetic field by two layers of
magnetic shielding. The detected far field diffraction patterns are shown in Fig. 2.
Figure 2(a), (b), (c), and (d) are taken at 1000 eV, 500 eV, 250 eV, and 125 eV,
respectively. They are taken by sweeping the diffraction pattern across the detection slit
using deflection plates, while electron counts were collected with a multi channel scaler
board. No background has been subtracted.
In a separate experiment, a similar metal coated grating was placed in an Hitachi S2460N Scanning Electron Microscope (SEM) after the objective lens. The SEM
pictures show evidence of electron beam diffraction for energies ranging from 500 eV
to 25 keV. This demonstrates that the grating can be used in conjunction with existing
electron microscope technology, and can tolerate 1nA mm2 . We failed to observe a
diffraction pattern from the unmetallized silicon nitride gratings in either apparatus,
presumably because of surface charge.
A path integral calculation without any interaction between the electron and the
grating does not agree well with the data, especially at the lower energies. Neither the
peak widths nor the relative peak heights agree well. Only the peak positions are
matched (Fig. 3, red dashed line). However, a modified analysis that includes a
spatially dependent potential energy, V(x), to describe the interaction between the
electron and the grating channel walls, does improve the agreement to the diffraction
data (Eq. (1). Two types of potential V(x) are then considered. One potential based on
the method of images for an electron next to a conducting wall (Eq. 2) and an
additional, random potential (Eq. 3), that appears to be needed to explain the broadening
of the diffraction peaks.
5
  Ne
i path
;
Lpath
 path 
dB
path
Vimage ( x) 
qimage
4 0  w  2 x 

2 
qV ( x)t
qimage
4 0  w  2 x 
(1)
(2)
The potential due to the image charge (Eq. 2) improves the relative diffraction peak
ratio considerably (Fig. 3, blue line). This potential depends on the distance of the
electron from the grating channel wall which is approximated by an infinite plane in our
calculation. The total width of the channel between the two grating bar walls is denoted
by w.
For curves in Fig. 3 the value of qimage that worked best was -e/3.6.
In addition to the modified peak intensities, we observed an additional widening of
the diffracted peaks relative to the raw beam (without the grating). Rotating the grating
about the beam axis by 90o (so the bars are parallel to the axis of translation of the
detector) does not reduce the excess width. To explain this anomaly we added an
irregular potential, made from a series of Gaussians,
Vrandom ( x)   Ae
i
4ln 2 x  xi 
i
2
2
(3)
i
to the theoretical model of the grating (Figs. 2 and 3, black lines). In Eq. (3), Ai and i
are obtained from random number generators. The random numbers Ai and i are
normally distributed with means of 0 eV and 250 nm and FWHM of 0.4 eV and 250
nm, respectively. The quantity xi gives us a new center for each subsequent Gaussian
which has been shifted from the previous Gaussian by 2i. This potential has a
characteristic lateral structure of 250 nm. Incoherent averaging of multiple diffraction
patterns for different random potentials effectively reduces the transverse coherence
length for electrons at the grating to approximately 250 nm, and hence broadens the
transmitted beam and diffraction peaks. This decoherence process is enigmatic given
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that the grating bars are only 50 nm wide. It is not clear, for example, if the random
potentials change in time.
The interaction of electrons with grating bars is potentially strong enough to cause this
experiment to fail, either because of image charges or random charges on the bar
surfaces. Using a rough estimate, the image charge potential of a stationary electron in
a 50 nm wide slot between infinite conducting plates is approximately 0.06 eV in the
middle. At 125 eV the transit time is 1.9 1014 s , hence the image potential for a static
electron would cause a phase shift of 1.7 radians in the middle and a larger phase shift
toward either side of the channel between the bars. This phase curvature in each
channel is found to change the strength of each diffraction order, but it does not broaden
them.
The image charge has an even larger effect if the grating is used at non-normal
incidence [11]. In Fig. 4 (top) the electron transmission into the zeroth diffraction order
is given (square dots) as a function of the grating angle. The dashed line is the result of
diffraction theory with no interaction between the electron and the grating bars. The
dash dot line is the result of the same calculation with a potential added that describes
the image charge interaction between a static electron and two infinite walls, Eq. (2)
[12, and references there in]. The solid line is a similar calculation with the image
potential reduced by a factor of 3.6 in strength. Figure 4 (bottom) shows similar
measurements and calculations for the electron diffraction into the first order. The
asymmetry in the model is due to the cross-sectional geometry of the grating bars
combined with the image charge [11], and the data appear consistent with this model.
In summary, free-standing nanostructure transmission gratings have the advantage
that they work at low energies (< 1 keV) and we expect them to enable new electron
interferometry and holography experiments. The data shown in Fig. 2 were obtained
7
with a diffraction-limited electron beam; however, much less collimation is required to
observe diffraction, and even less collimation is needed for Talbot-vonLau (near field)
interferometry in which two well-defined momentum states are created in the same
spatial region. Hence, we claim that nanostructured gratings can simultaneously play
the role of the coherent source and beam splitter in electron interferometry.
The broadening of the diffraction peaks within our model suggests a new kind of
decoherence mechanism that may reduce interference fringe contrast. For one of the
worst case scenarios we estimated this effect. An interferometer may use a second
grating to recombine two diffracted beams. The two diffracted beams pass through
different parts of the second grating and thus accumulate a different spatially varying
phase. Our model based on random potentials allows us to predict the resulting
interferometric contrast when these two beams are once again overlapped. Using a
detection slit of 150 m we find a contrast of 27% and 15% for 500 and 125 eV
respectively, while for a detection slit of 10 m we find a contrast of 64% and 35% for
500 and 125 eV. So there is encouraging evidence that low energy electron
interferometry may be feasible with these gratings.
Applications for low-energy electron interferometers include measurements of the index
of refraction due to forward scattering from atoms [13], sensing vector [14-17] and
scalar potentials [18]. High efficiency diffraction gratings for low energy electrons can
also improve the performance of off-axis electron holography microscopy [19].
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Wave—particle duality of C60 molecules. Nature 401, 680 (1999).
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3.2, low energy electron ranges in matter 2001, http://xdb.lbl.gov/.
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Heisenberg uncertainty principle for fullerene molecules. Phys. Rev. A. 65, 03219
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material grating. Phys. Rev. A 70, 043607 (2004).
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Cim. 110A, 571 (1997).
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[13] Forrey, R.C., Dalgarno, A., and Schmiedmayer, J., Determining the electron
forward-scattering amplitude using electron interferometry, Phys. Rev. A 59, R942
(1999).
[14] Aharonov, Y. and Bohm, D., Significance of electromagnetic potentials in quantum
theory. Phys. Rev. 115, 485 (1959).
[15] Boyer, T.H. Semiclassical explanation of the Matteucci-Pozzi and AharonovBohm phase shifts. Found. Phys. 32, 41 (2002).
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theory. Phys. Rev. Lett. 92, 070404 (2004).
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topological nonlocal quantum interference effects. Phys. Rev. Lett. 92, 020401 (2004).
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Acknowledgements
We thank Hank Smith for the production of the gratings in his laboratory. We thank
Mike Chapman for helpful discussions. We also thank Sy-Hwang Liou for the use of
his focused ion beam to make a collimation slit. This material is based upon work
supported by the National Science Foundation under Grant No. 0112578. This material
is also based upon work supported by the Department of the Army under Grant No.
DAAD1902-1-0280, and the content of the information does not necessarily reflect the
position or the policy of the federal government, and no official endorsement should be
inferred.
Correspondence and requests for materials should be addressed to H.B. (e-mail:
hbatelaan2@unl.edu).
10
100 nm
FIG. 1. Nano-fabricated 100-nm-period grating.
11
(a)
6000
20000
15000
Count rate
4000
10000
2000
5000
0
-400
-200
0
200
0
-200
400
0
200
0
200
0
200
0
200
(b)
4500
7000
3750
Count rate
5250
3000
2250
3500
1500
1750
750
0
-400
(c)
-200
0
200
0
-200
400
3500
2250
3000
Count rate
2500
1500
2000
1500
750
1000
500
0
-400
Count rate
(d)
-200
0
200
0
-200
400
1500
1800
1000
1200
500
600
0
-400
-200
0
200
400
0
-200
Position(m)
FIG. 2. Electron beam diffraction. The electron diffraction is presented as a function of
position for electron energies of (a) 1000 eV, (b) 500 eV, (c) 250 eV, and (d) 125 eV.
The beam profile without a diffraction grating is shown for comparison. The result of a
path integral calculation is represented by the solid line.
12
1500
Count rate
1000
500
0
-400 -200
0
200
400
Position(m)
FIG. 3. Diffraction models for 125 eV . Data points are identical to figure 2 d. Red
dashed line is the result of a path integral calculation without an image charge or a
random potential. Blue solid line is the result of a path integral calculation with image
charge potential. Black solid line includes both random and reduced image potentials.
13
25000
Count Rate
20000
15000
10000
5000
0
12000
-30
-20
-10
0
10
20
30
0
10
20
30
10000
Count Rate
8000
6000
4000
2000
0
-30
-20
-10
Angle(degrees)
FIG. 4. Rocking curves for 500 eV . Top: Electron transmission in the 0-order is given
as a function of grating angle. Bottom: Electron transmission in the 1st order is given as
a function of grating angle (for explanation see text).