1

Free-standing, metal-coated transmission gratings for electrons

Glen Gronniger1, Brett Barwick1, Tim Savas2, Alex Cronin3, Dave Pritchard2, and

Herman Batelaan1

1Department 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 low-

energy 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

. This agrees with the observed angular spread for our

500 eV electron beam which produces a 7 m wide spot (FWHM) in the plane of

detection, located 31 cm downstream of the 2 slit. The geometrical transverse

momentum spread through the second slit is . The

corresponding transverse coherence length is [10], which means

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 , and

coherent over (at least) 2 . 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

periodicity. The coherence length is given by (at 500 eV,

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

due to the energy width of the electron beam, , where n

is the diffraction order, is negligible (the associated longitudinal coherence length and

time are about 50 nm and ). 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 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 S-

2460N 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 . 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

(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,

(3)

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 2i. 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

6

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 , 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].

8

[1] Keith, D.W., Schattenburg, M.L., Smith, H.I., and Pritchard, D. Diffraction of

atoms by a transmission grating. Phys. Rev. Lett. 61, 1580 (1988).

[2] Savas, T.A., Shah S.N., Schattenburg, M.L., Carter, J.M., and Smith, H.I..

Achromatic interferometric lithography for 100-nm-period gratings and grids. J. Vac.

Sci. Technol. B 13(6), 2732 (1995).

[3] Keith, D.W., Ekstrom C.R., Turchette, Q.A., and Pritchard, D.E. An interferometer

for atoms. Phys. Rev. Lett. 66, 2693 (1991).

[4] Schollkopf, W. and Toennies, J.P.. The nondestructive detection of the helium

dimer and trimer. J. Chem. Phys. 104, 3 (1996).

[5] Arndt, M., Nairz, O., Vos-Andreae, J., Keller, C., van der Zouw, G., Zeilinger, A..

Wave—particle duality of C60 molecules. Nature 401, 680 (1999).

[6] Clauser, J.F., and Li, S. Talbot-vonLau atom interferometry with cold slow

potassium. Phys. Rev. A. 49, R2213 (1994).

[7] Holl, V.P. Ein elektronenoptischer Mikroschreiber hoher Stabilität und seine

Anwendung. Optik 30, 116 (1969).

[8] Pianetta, P., X-ray Data Booklet, Lawrence Berkeley National Laboratory, section

3.2, low energy electron ranges in matter 2001, http://xdb.lbl.gov/.

[9] Jönsson, C. Electron diffraction at multiple slits. Am. J. Phys. 42, 4 (1974).

[10] Nairz, O., Arndt, M., and Zeilinger, A. Experimental verification of the

Heisenberg uncertainty principle for fullerene molecules. Phys. Rev. A. 65, 03219

(2002).

[11] Cronin, A.E. and Perreault, J.D. Phasor analysis of atom diffraction from a rotated

material grating. Phys. Rev. A 70, 043607 (2004).

[12] Ross, D.K. Interferometric test of the electron mass shift in a cavity. IL Nuovo

Cim. 110A, 571 (1997).

9

[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 Aharonov-

Bohm phase shifts. Found. Phys. 32, 41 (2002).

[16] Aharonov, Y. and Kaufherr, T. The effect of a magnetic flux line in quantum

theory. Phys. Rev. Lett. 92, 070404 (2004).

[17] Aharonov, Y., Popescu, S., Reznik, B., and Stern, A. Classical analog to

topological nonlocal quantum interference effects. Phys. Rev. Lett. 92, 020401 (2004).

[18] Matteucci, G. and Pozzi, G. New diffraction experiment on the electrostatic

Aharonov Bohm effect. Phys. Rev. Lett. 54, 2469 (1985).

[19] Ru, Q. Incoherent electron holography. J. Appl. Phys. 77, 1421 (1995).

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

FIG. 1. Nano-fabricated 100-nm-period grating.

100 nm

11

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

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

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).