^{1}, Glen Gronniger

^{1}, Lu Yuan

^{1}, Sy-Hwang Liou

^{1}and Herman Batelaan

^{1,a)}

### Abstract

Electron diffraction from metalcoated freestanding nanofabricated gratings is presented, with a quantitative path integral analysis of the electron-grating interactions. Electron diffraction out to the 20th order was observed indicating the high quality of our nanofabricated gratings. The electron beam is collimated to its diffraction limit with ion-milled material slits. Our path integral analysis is first tested against single slit electron diffraction, and then further expanded with the same theoretical approach to describe gratingdiffraction. Rotation of the grating with respect to the incident electron beam varies the effective distance between the electron and grating bars. This allows the measurement of the image charge potential between the electron and the grating bars. Image charge potentials that were about 15% of the value for that of a pure electron-metal wall interaction were found. We varied the electron energy from . The interaction time is of the order of typical metal image charge response times and in principle allows the investigation of image charge formation. In addition to the image charge interaction there is a dephasing process reducing the transverse coherence length of the electron wave. The dephasing process causes broadening of the diffraction peaks and is consistent with a model that ascribes the dephasing process to microscopic contact potentials. Surface structures with length scales of about observed with a scanning tunneling microscope, and dephasing interaction strength typical of contact potentials of support this claim. Such a dephasing model motivated the investigation of different metallic coatings, in particular Ni, Ti, Al, and different thickness Au–Pd coatings. Improved quality of diffraction patterns was found for Ni. This coating made electron diffraction possible at energies as low as . This energy was limited by our electron gun design. These results are particularly relevant for the use of these gratings as coherent beam splitters in low energy electron interferometry.

The authors thank Tim Savas for production of the gratings. This material is based upon work supported by the National Science Foundation under Grant No. 0112578.

I. INTRODUCTION

II. THEORY

A. Theory of single slit electron diffraction

B. Theory of multiple slit electron diffraction

C. Image charge potential

D. Random potential

E. Grating tilt dependence of potentials

III. EXPERIMENTAL SETUP

IV. RESULTS

A. Single slit diffraction

B. Electron diffraction from gratingscoated with

C. Image charge potential measurement

D. Random potentials: Electron diffraction from gratings with different coatings

E. Grating quality

F. Lowest energydiffraction patterns

V. SUMMARY AND DISCUSSION

### Key Topics

- Diffraction gratings
- 142.0
- Metallic coatings
- 36.0
- Electron diffraction
- 28.0
- Electron beams
- 22.0
- Interferometers
- 22.0

## Figures

Nanofabricated period grating. Figure courtesy of Tim Savas.

Nanofabricated period grating. Figure courtesy of Tim Savas.

Geometry of the single slit theory. There are three axes, with depicting the incoherent slit plane, depicting the single slit plane, and depicting the detection plane. The distances between the planes are and . The dashed line is the zero point for the , , and variables. The dotted lines show a few of the possible straight paths that the electron can take. All phases that are accumulated along these paths are stored in the kernel. The position of the initial delta function indicates the location of the incoherent source.

Geometry of the single slit theory. There are three axes, with depicting the incoherent slit plane, depicting the single slit plane, and depicting the detection plane. The distances between the planes are and . The dashed line is the zero point for the , , and variables. The dotted lines show a few of the possible straight paths that the electron can take. All phases that are accumulated along these paths are stored in the kernel. The position of the initial delta function indicates the location of the incoherent source.

Cross sectional cut of the nanofabricated grating. The bevel angle of the slit is given by , the electron beam angle with respect to the grating is , the slit width is given as , and the grating thickness is given by .

Cross sectional cut of the nanofabricated grating. The bevel angle of the slit is given by , the electron beam angle with respect to the grating is , the slit width is given as , and the grating thickness is given by .

Illustration of the experimental setup. The electrons are created in the electron gun using a tungsten filament. The filament is floated at a negative voltage and the electrons are focused using several lenses. After leaving the source the electrons are collimated by 5 and slits. After traveling through the two collimation slits the electrons pass through a metallic coated nanofabricated grating, which can be moved into the beam with a vertical linear feedthrough. After passing through the diffraction grating the diffraction pattern is scanned by either moving a detection slit across the pattern or by rastering the electron beam with deflection plates. The electrons are then detected with an electron multiplier placed behind the detection slit.

Illustration of the experimental setup. The electrons are created in the electron gun using a tungsten filament. The filament is floated at a negative voltage and the electrons are focused using several lenses. After leaving the source the electrons are collimated by 5 and slits. After traveling through the two collimation slits the electrons pass through a metallic coated nanofabricated grating, which can be moved into the beam with a vertical linear feedthrough. After passing through the diffraction grating the diffraction pattern is scanned by either moving a detection slit across the pattern or by rastering the electron beam with deflection plates. The electrons are then detected with an electron multiplier placed behind the detection slit.

single slit diffraction, taken at . The dots represent the experimental data, and the solid lines represent the results of the path integral model including incoherent averaging over the first slit and convolution over the detection slit. The dashed line represents the Fraunhofer optical diffraction theory with a plane wave illumination of the single slit. The theory is normalized to the experimental data.

single slit diffraction, taken at . The dots represent the experimental data, and the solid lines represent the results of the path integral model including incoherent averaging over the first slit and convolution over the detection slit. The dashed line represents the Fraunhofer optical diffraction theory with a plane wave illumination of the single slit. The theory is normalized to the experimental data.

single slit diffraction, taken at . The dots represent the experimental data, and the solid lines represent the results of the path integral model including incoherent averaging over the first slit and convolution over the detection slit. The dashed line represents the Fraunhofer optical diffraction theory with a plane wave illumination of the single slit. The theory is normalized to the experimental data.

(Color online) Electron diffraction (a) at 900 and (b) . The diffraction patterns are on the left and the beam profiles with the gratings removed are on the right. The solid line is the path integral model with random potential, image charge, and incoherent source.

(Color online) Electron diffraction (a) at 900 and (b) . The diffraction patterns are on the left and the beam profiles with the gratings removed are on the right. The solid line is the path integral model with random potential, image charge, and incoherent source.

Diffraction patterns taken at of grating tilt. Graph (a) is for , (b) is for , (c) is for , and (d) is for . These diffraction patterns taken at have approximately the largest asymmetries of all the tilt angles.

Diffraction patterns taken at of grating tilt. Graph (a) is for , (b) is for , (c) is for , and (d) is for . These diffraction patterns taken at have approximately the largest asymmetries of all the tilt angles.

(Color online) Rocking curves for . The electron transmission into a particular order is given as a function of grating tilt angle. Graph (a) is zeroth order, (b) is the first order, (c) is the second order, (d) is the third order, and (e) is the fourth order. The squares and circles are the +/− orders. The solid line is the model with image charge potential, random potential, and geometry of the grating included.

(Color online) Rocking curves for . The electron transmission into a particular order is given as a function of grating tilt angle. Graph (a) is zeroth order, (b) is the first order, (c) is the second order, (d) is the third order, and (e) is the fourth order. The squares and circles are the +/− orders. The solid line is the model with image charge potential, random potential, and geometry of the grating included.

(Color online) Rocking curves for . The electron transmission into a particular order is given as a function of grating tilt angle. Graph (a) is zeroth order, (b) is the first order, (c) is the second order, (d) is the third order, and (e) is the fourth order. The squares and circles are the +/− orders. The solid line is the model with image charge potential, random potential, and geometry of the grating included.

(Color online) Rocking curves for third order. The electron transmission into the third order is given as a function of grating tilt angle. The solid circles are data. The solid line is the model with , the dotted line is with , and the dashed line is with .

(Color online) Rocking curves for third order. The electron transmission into the third order is given as a function of grating tilt angle. The solid circles are data. The solid line is the model with , the dotted line is with , and the dashed line is with .

(Color online) Diffraction patterns at . The different coatings are (a) deposition of , (b) deposition of , (c) Ti, and (d) Ni. The open squares are the diffraction pattern, and the solid dots are the associated beam data. The left vertical scales are for the diffraction pattern counts and the right vertical scales are for the beam counts.

(Color online) Diffraction patterns at . The different coatings are (a) deposition of , (b) deposition of , (c) Ti, and (d) Ni. The open squares are the diffraction pattern, and the solid dots are the associated beam data. The left vertical scales are for the diffraction pattern counts and the right vertical scales are for the beam counts.

(Color online) STM picture of the coated substrate. The lighter regions are protrusions on the substrate.

(Color online) STM picture of the coated substrate. The lighter regions are protrusions on the substrate.

Electron diffraction at . Both plus and minus orders are shown out to approximately 20th order.

Electron diffraction at . Both plus and minus orders are shown out to approximately 20th order.

Electron beam diffraction. Electron diffraction from Ni coated gratings presented as a function of position for energies of (a) 125 and (b) . The beam profile without a diffraction grating is shown for comparison.

Electron beam diffraction. Electron diffraction from Ni coated gratings presented as a function of position for energies of (a) 125 and (b) . The beam profile without a diffraction grating is shown for comparison.

## Tables

Broadening factors (BFs) as a function of different metallic coatings, at electron energy. The broadening factor is defined as , the ratio of the full width at half maximum of the diffraction peaks , over the full width at half maximum of the beam . The symbol denotes the open fraction of the grating which is defined as the slit width divided by the periodicity , where is .

Broadening factors (BFs) as a function of different metallic coatings, at electron energy. The broadening factor is defined as , the ratio of the full width at half maximum of the diffraction peaks , over the full width at half maximum of the beam . The symbol denotes the open fraction of the grating which is defined as the slit width divided by the periodicity , where is .

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