Diffractive electron mirror for use in quantum electron microscopy

(1)

Diffractive Electron Mirror for Use in Quantum Electron Microscopy by

Navid Abedzadeh

B.ASc. Nanotechnology Engineering University of Waterloo (2015)

Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Electrical Engineering and Computer Science at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2018

Signature of Author: . . . . Department of Electrical Engineering and Computer Science

January 31, 2018

Certified by . . . . Karl K. Berggren Professor of Electrical Engineering and Computer Science Thesis Supervisor

Accepted by . . . . Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students

(2)

(3)

Diffractive Electron Mirror for Use in Quantum Electron Microscopy by

Navid Abedzadeh

Submitted to the Department of Electrical Engineering and Computer Science on January 31, 2018 in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Electrical Engineering and Computer Science

ABSTRACT

Periodic atomic structures in thin crystals and artificially fabricated periodic structures in transmission gratings have long been used to coherently split electrons by means of electron diffraction for applications such as interferometry, holography and imaging. Due to their reliance on transmission through matter, however, these methods are prone to electron scattering and absorption and are therefore lossy to some extent. This loss becomes a major issue for quantum electron microscopy (QEM), an interaction-free mea-surement scheme with electrons as probe particles. QEM relies on single electrons completing many round trips inside an electron resonant cavity, splitting and re-coupling during each round trip, effectively mul-tiplying the probability of loss by the number of round trips. Thus, in one of the designs for QEM, the use of reflective diffraction gratings as lossless electron beam splitters is proposed. In this thesis, diffractive electron mirrors were fabricated by integrating one-dimensional diffraction gratings with tetrode electron mirrors. Optical interference lithography was used to fabricated silicon diffraction gratings with pitches varying from 200 nm to 500 nm. Furthermore, a proof-of-principle experiment to demonstrate their func-tion as electron mirrors inside a scanning electron microscope was developed. It was demonstrated that the constructed tetrode electron mirrors satisfied the requirements of QEM for electron energies up to 3 keV. Finally, in a similar experiment, the fabricated diffractive electron mirrors were tested to demonstrate their function as lossless beam splitters. Preliminary results point to the evidence for electron diffraction, suggesting that diffractive electron mirrors could be used as as lossless electron beam splitters for QEM and other applications.

Thesis Supervisor: Karl K. Berggren

(4)

(5)

Acknowledgments

The work presented in this thesis would not have been possible without constant support and advice from my research supervisor, Prof. Karl K. Berggren. Karl has created a friendly and open research environment in which I find myself able to remain inquisitive and autonomous while knowing that his office is two doors down from mine in case I need his support. I sincerely appreciate his guidance.

I would also like to thank Dr. Chung-Soo Kim who taught me most of what I know about mechanical assembly and electron optics associated with the experiments presented in this thesis. I shall never forget Chung-Soo’s patience with me even when I asked him less-than-intelligent questions.

In addition, I would like to thank Dr. Richard Hobbs, the first person who got me excited about this project and continued to be an incredible mentor to me. Also from our group, I would like to thank Akshay Agarwal and Marco Turchetti for many valuable discussions.

I would like to thank our collaborators at TU Delft, Maurice Krielaart and Prof. Pieter Kruit. Dis-cussions with Maurice provided great help in the writing Chapter 3 on the theory of electron diffraction. Pieter’s immense knowledge of electron optical systems and his valuable lecture notes on charged particle optics were significant resources in the writing of this thesis. I often learned more from a five-minute long conversation with Pieter than weeks of literature search on my own.

I would like to thank Dr. Timothy (Tim) Savas and James (Jim) Daley from the NanoStructures Labo-ratory at MIT. Tim trained me to use the Lloyd’s mirror and helped me with every step of the fabrication process. Jim’s constant support in the clean room made the fabrication process go smoothly. He also helped me with electron-beam evaporation in preparation of my samples.

I thank Gordon and Betty Moore Foundation for supporting the Quantum Electron Microscope project. I thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for partial funding of my graduate studies.

I would like to thank my parents. Ten years ago, in the hope of giving my sister and me a life in a more open society with better educational prospects, they gave up on their comfort, chose the arduous path of immigration and brought us to Canada. Without their sacrifice, none of my achievements would have been possible.

Finally, I would like to thank my girlfriend, Rana, who has been my biggest supporter throughout my academic career and has never lost faith in me. I thank her for her patience and kindness.

(6)

List of Figures

1.1 Schematic of a low-energy electron microscope . . . 12

1.2 Elitzur-Vaidman interaction-free measurement . . . 15

1.3 High-efficiency interaction-free measurement . . . 17

1.4 Interaction-free measurement inside a resonant cavity . . . 18

1.5 A design for a quantum electron microscope . . . 21

2.1 Two-aperture electron lens simulation . . . 23

2.2 Accelerating einzel lens simulation . . . 24

2.3 Flat electron mirror simulation . . . 26

2.4 Tetrode electron mirror simulation . . . 27

3.1 Electron diffraction by a transmission grating . . . 29

3.2 Different regions of a flat electron mirror . . . 30

3.3 Behavior of Airy functions . . . 32

3.4 Probability density function of electron near a mirror . . . 33

3.5 Simulation of equipotential surfaces near a grating . . . 33

3.6 Flat mirror with sinusoidal perturbation . . . 34

3.7 Design of a proof-of-principle experiment . . . 36

3.8 Flat tetrode mirror simulation . . . 38

3.9 Electron round trip trajectory simulation . . . 39

3.10 Electron trajectory mimicking diffraction . . . 40

4.1 Fabrication overview . . . 44

4.2 Lloyd’s mirror setup . . . 44

4.3 Micrograph of Grating after resist development . . . 46

(7)

4.5 Final fabrication results . . . 47

4.6 Diffractive electron mirror components and assembly . . . 48

5.1 micrograph of tetrode mirror results . . . 50

5.2 Micrograph of alignment process . . . 51

5.3 Resolution analysis of a reflected image . . . 51

5.4 Reflection without point symmetry . . . 52

5.5 Ray optics analysis (MATLAB simulation) . . . 53

5.6 Significance of pivot point imaging . . . 53

5.7 Evidence for electron diffraction . . . 54

5.8 Effects of objective lens aberration on diffraction spots . . . 56

5.9 Image separation as a function of immersion lens strength . . . 57

5.10 Comparison between experimental and processed images . . . 58

6.1 Diffractive electron mirrors in low-dose microscopy . . . 61

6.2 Electron mirror aberration correction . . . 63

6.3 Spherical aberration diagnostics with shadow imaging . . . 63

(8)

Chapter 1 Avoiding radiation damage in electron

microscopy

The ability to image biomolecules with atomic-scale resolution is key to understanding biological systems and processes. Due to diffraction, the smallest resolvable feature size achieved by optical microscopy is often too large for imaging of biological molecules such as DNA, RNA or proteins. This fundamental limit is directly proportional to the wavelength of the probing particles–photons in case of optical microscopy. Since the de Broglie wavelength of electrons at readily achievable energies is much smaller than that of photons, electron microscopy enables us to push the diffraction limit to far smaller length scales. Therefore, electron microscopy is the immediate candidate when it comes to resolving objects on an atomic-scale resolution. Scanning electron microscopes (SEM) are routinely used to achieve nanometer (nm) resolution while transmission electron microscopes (TEM) and scanning transmission electron microscopes (STEM) have been shown to resolve down to angstrom length scales [1, 2, 3, 4].

Although these tools have revolutionized our ability to peer into and engineer matter on an atomic level, imaging of biological samples remains a challenge [5]. The difficulty arises not from how small we can make the de Broglie wavelength but from the fact that energetic electrons can easily damage organic samples. Therefore, the specimen very often will be destroyed before an adequate image with atomic resolution could be produced. Many different damage mechanism such as heating, electrostatic charging, radiolysis, displacement damage and sputtering have been identified and studied [6]. It has been shown that two important variables determining the extent of radiation damage are the energy and the dose of probing electrons [7]. The dose impact goes as one would expect: the higher the dose, the more damage the sample will sustain. However, there are a few subtleties with regards to the energy which will be discussed

(11)

in this chapter.

To mitigate beam-induced damage, several approaches have been proposed, each with their own ad-vantages and disadad-vantages. Among these approaches are low-energy schemes, low-dose schemes and schemes employing principles of quantum mechanics.

1.1 Low-energy schemes

High-energy probe electrons are recognized as the source of radiation damage to biological samples and as such, a number of schemes have been proposed and implemented to use lower-energy electrons in order to lower radiation damage. Among these approaches are low-energy electron microscopy (LEEM) [8, 9, 10] and low-energy electron in-line holography [11, 12].

In LEEM, the sample is placed directly on top of and in physical contact with an electrostatic immer-sion lens. The immerimmer-sion lens, also known as cathode lens, behaves as an electron mirror when a potential equivalent to the energy of the incident electron is applied to it. Hence, by controlling the negative poten-tial the electrode and the sample are subject to, one could control the landing energy of electrons on the sample. For instance, if the cathode and the sample were biased to −999 V, a 1 keV electron beam would decelerate to only 1 eV and gently land on the sample. The backscattered electrons are then accelerated away from the sample and used to produce an image of the sample. Figure 1.1 shows a schematic of a low-energy electron microscope.

In practice, the landing energy in LEEM is kept within 0-100 eV to stay below energy thresholds for different damage mechanisms for various samples. The ionization energy of C−H bonds, prominent in all biological samples, is 4.2 eV [13]. At this electron energy, the de Broglie wavelength is 0.6 nm. However, taking other effects such as optics, specimen characterization and current requirements into account, the theoretical resolution of LEEM is estimated to be in the range of 2-4 nm [13, 14].

Low-energy electron in-line holography also takes advantage of low-energy electrons (20-300 eV) to reduce radiation damage [15, 12]. In this scheme, a suspended biological sample could be placed directly in front an atomically sharp electron source tip such that part of a coherent electron beam could transmit through the sample while the rest could go around the sample undisturbed. If a screen is placed a certain distance away from the sample, due to the interference between the two parts of the beam, a holographic image of the sample could be recorded on the screen. Magnification occurs merely based on the geometry of the setup: sample-screen distance determines the magnification for a given tip-sample distance. A significant advantage of this technique is the lack of a need for an objective lens and hence the absence of

(12)

Figure 1.1: Schematic of a low-energy electron microscope (LEEM) system. Negatively biased sample causes the incident electron to decelerate and land with a lower energy (0 − 100 eV) to reduce radiation damage due to high-energy electron beams. Figure taken from R.M. Tromp [10]

all aberrations associated with electron lenses.

The aforementioned low-energy schemes are reported to reduce radiation damage to specific biological samples. Meanwhile, it has also been observed that even ultra low-energy electrons (0-4 eV) could, in sufficient doses, cause single strand breaks in DNA molecules [16]. The mechanism for such damage is referred to as dissociative electron attachment which involves short-lived core-excited anion states of otherwise excited yet neutral atoms in the sample [17]. Another important consideration in microscopy with ultra low-energy electrons is that at a certain point, the gains from small de Broglie wavelengths of electrons begin to dissipate.

1.2 Low-dose schemes

Due to the quantized nature of electrons, image information in conventional electron microscopy is shot-noise limited. Shot-shot-noise could be attributed to the Poissonian statistics of electron sources and detectors. In practice, this noise appears as randomly distributed black and white spots on a micrograph, restricting the amount of useful information that the image could hold. To mitigate the negative effects of

(13)

shot-noise, microscopists have relied on longer averaging times and using a larger number of probe particles (larger beam currents) to remove some of the uncertainty due to the Poissonian processes. These solutions, however, are not always applicable. In the case of biological samples, one cannot use an arbitrarily large dose of electrons for imaging since the sample will often be altered, damaged or destroyed long before an image with an acceptable signal-to-noise ratio (SNR) is formed. Hence, shot-noise introduces an effective resolution limit: to ensure minimal damage to biological samples, one is forced to sacrifice resolution [18]. Biological samples, made up of low-atomic-number elements, scatter high energy electrons minimally, often only once. Such specimens are referred to as weak-phase objects since the phase shift they impart on the transmitted electrons is very small [19]. Phase contrast imaging, turning electrons’ phase informa-tion into amplitude contrast, is known to be the most dose-efficient conveninforma-tional technique for imaging weak-phase objects [20, 21]. Phase contrast imaging in electron microscopy is achieved in conventional TEM by deliberately introducing some defocus or other aberrations or by the use of phase plates [22] or through in-line [11] or off-axis holography [23]. Ptychography is another technique that has been shown to achieve relatively dose-efficient phase contrast imaging by taking advantage of phase information in the overlapping diffraction disks formed in STEM [21]. However, although these techniques achieve better dose-efficiencies (∼ 106_e−_/nm2_{) [21] compared to conventional amplitude-contrast imaging, to prevent}

damage to biological samples, electron dose must be limited to ∼ 103_e−_/_nm2_{[24]. At such low beam}

cur-rents, current phase contrast imaging techniques are not capable of producing acceptable image contrast for weak-phased objects such as biomolecules.

Cryo-electron microscopy (cryo-EM) avoids this problem by using a large ensemble of identical sam-ples instead of imaging a single sample, effectively spreading the risk of damage over a large number of samples. Thousands of images of these identical samples can then be averaged during post-processing to achieve an apparently high-SNR images [25]. As impressive as the images acquired by cryo-EM are [26], there are a number of drawbacks to this technique. The most obvious challenge is the need for thousands of identical copies of the same object. This need limits the sample pool of appropriate biological samples as not all biomolecules could be prepared with thousands of their exact replicas within the same speci-men. Furthermore, due to averaging, conformational information of the individual particle is lost [25, 15]. Most importantly, cryo-EM does not address the fundamental problem: radiation damage to an individual biomolecule [18].

(14)

1.3 Quantum mechanical solutions

As severe an obstacle as shot-noise is in microscopy, it is not a fundamental limit of measurement deep-rooted in the laws of physics. Shot-noise is merely a consequence of the employed classical detection strategy [27]. By taking advantage of the principles of quantum mechanics such as the use of squeezed photon states or quantum correlation between input photons, sub-shot-noise measurements have been demonstrated [28, 29]. Although these advancements in quantum measurement have opened up new exciting avenues for accurate and efficient measurements using photons as probe particles, their promise for use in electron microscopy has been largely unfulfilled. This nonfulfillment could be attributed to the fact that preparation of electrons with higher-than-classical correlation in free space has not been realized [30].

Multi-pass microscopy, already demonstrated to beat the shot-noise limit in optical microscopy [31], is one avenue which holds promise for sub-shot-noise imaging in electron microscopy [32]. Multi-pass microscopy is a phase contrast imaging scheme that could achieve similar gains in SNR as entangled-probe-particle measurements do for a given damage, without a need for entangled particles. In other words, electron microscopy could take advantage of this scheme to gain more useful information out of phase-contrast imaging: beating the shot-noise limit.

The focus of this section, however, will be on another quantum mechanical scheme known as interaction-free measurement (IFM) and the attempts to realize such a technique in an electron microscope.

1.3.1 Basic interaction-free measurement

Elitzur and Vaidman (EV) in 1993 proposed a thought experiment for a measurement scheme that could detect the presence or absence of a particle without directly interacting with it [33]. To setup the problem, they assumed a Mach-Zehnder interferometer similar to the one shown in Fig. 1.2. It is possible, by setting up the interferometer’s arm lengths just right, to ensure that photons entering the system from the left, will always, with 100% certainty, be detected by detector D1 (bright port). This task is done by

taking advantage of the wave nature of photons at the second beam-splitter; namely, due to destructive interference at detector D2 (dark port), the photons will always emerge at D1. At this point, if an opaque

sample is placed in one of the arms of the interferometer as shown in Fig. 1.2, three possible outcomes could occur:

(i) no detector clicks (ii) detector D2clicks

(15)

(iii) detector D1clicks

Now let us go through each of these three possible outcomes one by one. First, consider the case of no detector clicking which implies that the input photon was absorbed (or scattered) by the opaque sample. This outcome would occur 50% of the times–when the input photon is transmitted, rather than reflected at the first 50-50 beam splitter. Therefore, the case of no click could be thought of as interaction with or damageto the sample.

D1

D2

Figure 1.2: Schematic of a Mach-Zehnder interferometer. Dashed lines represent 50-50 beam splitters; solid thick lines represent mirrors; dashed circle represents an opaque object. In the absence of an object, the optical path lengths have been set up such that a single photon entering the interferometer from the bottom left port would always end up at detector D1(bright port). The presence of an opaque object placed on one

of the arms of the interferometer could destroy the interference even when the photon is not absorbed by the object (detection at D2, dark port), enabling interaction-free measurement [33]

The authors, for dramatic effects, introduced the idea that the particle could be a photosensitive bomb which detonated if it absorbed a single photon. Moreover, there is a stock of such bombs with some being out of order (duds). Dud bombs miss a small part of their photodetectors such that the input photons could go through them without detonating them. The objective is to distinguish a functioning bomb from a dud without blowing it up!

Now, consider the case where detector D2clicks. Note that the initial setup ensured that in the absence

of any sample, all photons would end up at D1. Therefore, a click at D2indicates that not only there must

have been a sample in one of the arms of the interferometer, but also that the sample did not absorb the photon for if it had, the photon would not have caused a click at D2. Back to the bomb analogy. If D2clicks,

(16)

detonate, we can be further confident that the probe photon did not interact with it. This case constitutes as interaction-free measurement (IFM) which occurs 25% of the times in the setup shown in Fig. 1.2.

Finally, consider the case where detector D1clicks. Keeping in mind that the initial state of the

inter-ferometer was set up such that in the absence of a sample, D1always clicked, this outcome does not tell us

much. It could be that D1 clicks because there is no sample or that it clicks because the photon reflected

off of both beam splitters. In the bomb interrogation scenario, a click at D1does not distinguish between

a functioning and a dud bomb. This outcome occurs 25% of the times and when it does, another photon must be sent through the interferometer for further interrogation. It is important to note that each time the experiment is repeated, we introduce the risk of damage to the sample. More dramatically, a bomb could detonate!

Note that the wave-particle duality of a single quantum is at the heart of IFM: in the absence of an object, the wave nature of a single photon ensures, through destructive interference, that only the bright port (D1) clicks. However, in the presence of an object, it is the indivisibility of the single photon that

guarantees that in the event of a click at the dark port (D2), IFM has occurred [34].

1.3.2 High efficiency interaction-free measurement

If we consider beam splitters of arbitrary reflectivity R and transmissivity T = 1 − R, the original EV thought-experiment would yield a photon absorption event with probability of T = 1

2, IFM with

proba-bility of RT = 1

4, and inconclusive click at D1 with probability of R2 = 1

4. For an ideal Mach-Zehnder

interferometer with no loss, one could introduced a fraction η, representing the efficiency parameter, as follows [34]:

η = P (det)

P (det) + P (abs) (1.1)

where P (det) is the probability of detecting the presence of an object–IFM–and P (abs) is the probability that the single photon is absorbed by the object–damage. Therefore, for the original EV scheme, η =

RT RT +T =

R

R+1. It is obvious that the maximum fraction η → 1/2 is achieved when R → 1. Note that for a

single trial, a measurement is either interaction free, inconclusive or damaging; η is a measure of efficiency for repeated trials.

Two years after the original EV scheme, Kwiat et al. proposed a higher-efficiency IFM scheme [34]. They, instead of 50-50 beam splitters, proposed the use of beam splitters with reflectivity R = cos2_{(π/2N )}_,

and instead of a single Mach-Zehnder interferometer, introduced the idea of lining up N beam splitters in series as shown in Fig. 1.3a.

(17)

D1 D2 D1 D2

(a)

(b)

Figure 1.3: Mach-Zehnder interferometers in series for high-efficiency IFM. An input single photon enters the setup from the bottom left port and experiences N beam splitters of reflectivity R = cos2_{(π/2N )}

represented by gray rectangles. (a) In the absence of opaque objects in the upper arm of the system, the photon will go through a gradual transference of amplitude from the bottom arm to the upper one and will ultimately be detected by detector D1. (b) Opaque objects placed in the upper half prevent the photon

probability density function to build up on the upper arm and hence it will be detected at detector D2

without getting absorbed by the objects.

Initially, in the absence of an object, a single photon entering the system from the bottom left port will experience a small probability of transmission at each beam splitter. However, since the relative phases between the upper and lower paths are set to be zero, the amplitude of the photon begins a gradual trans-ference from the lower to the upper half of the interferometer system. This transtrans-ference could be thought of as gradual growth of the probability density of the photon in the upper half due to constructive inter-ference in the upper half and destructive interinter-ference in the lower half.

Kwiat et al. then considered opaque objects (photodetectors) placed in the upper half of the inter-ferometer system such that any transmitted photon would be absorbed by them (Fig. 1.3b). With R = cos2(π/2N )as the reflectivity of each beam splitter, the probability of detecting the photon at detector D2

at the bottom right port would be

P =hcos2 π 2N

iN

. (1.2)

The above probability entails that not a single photodetector in the upper path is triggered. Naturally, the complement of P is the probability that at least one photodetector was triggered (1 − P ). We can now set up the efficiency fraction for this arrangement:

η = P

P + (1 − P ) = P. (1.3)

At the limit of large N, P = 1 − π2_/4N_{. It can be seen that IFM with this scheme with N ≥ 4 already}

(18)

as one wishes; just increase N.

The authors recognized that the interferometer arrangement shown in Fig. 1.3 is not conducive to real-world experimentation; preparing many copies of a sample is not an intuitive task. Even having a single sample stretched over the entire upper branch is not practical. A more practical approach is to place a beam splitter of reflectivity R = cos2_{(π/2N )}_{in between two mirrors (resonant cavity) as shown in Fig.}

1.4. In this design, the left and right halves of the cavity could be thought of as the lower and upper halves of the interferometer system shown in Fig. 1.3, respectively. In this picture, the number of round trips that the photon goes through is equivalent to the number of beam-splitters N. Therefore, for high efficiency IFM, N must be large. Analogous to the case of the Mach-Zehnder interferometers in series, the cavity is initially set up such that there would be a gradual transfer of photon amplitude from the left to the right half after N round trips in the absence of an object. With an opaque object placed on the right side, the probability amplitude could not build up on the right side and hence the photon remains on the left side [34].

Figure 1.4: Schematic of high efficiency interaction-free measurement inside a cavity bounded by two mir-rors (M) and a beam splitter (BS) of reflectivity R = cos2 π

2N

. When an opaque object, a photosensitive bomb for dramatic effects, is placed on the right side of the cavity, the photon will be found on the left side of the cavity after a large number of round trips N, indicating the presence of the object without blowing it up! [34]

1.3.3 Quantum electron microscopy: an interaction-free measurement scheme with electrons

Extending the idea of interaction-free measurement (IFM) to electron microscopy was first proposed by Putnam and Yanik in 2009 [35]. At first glance, this task may seem like a mere change of probe particles. Al-though the physics of IFM as a single particle measurement technique remains largely the same, an attempt to experimentally achieve IFM with electrons would be considerably more complex. For instance, optical resonant cavities are well-established and could be set up with high quality mirrors and accurate control over optics’ distances. An electron cavity on the other hand is a far less trivial undertaking. The same could be said about beam splitters. Putnam and Yanik proposed the use of coupled electron ring guides

(19)

(as a cavity) with a guide potential that coupled the localized electron packet in a double-well potential (as a beam splitter/coupler). However, it was not until 2016 that more practical designs for incorporating interaction-free measurement into electron microscopy was proposed by Kruit et al [18]. These designs were published under the umbrella name: Quantum Electron Microscope (QEM)1_.

In this thesis, the emphasis will be placed only on one of these designs which involves creating a resonant electron cavity in the volume between two parallel electron mirrors and a mechanism by which to split and recouple the electrons. Henceforth, this specific design will be referred to as the QEM. Note that these components, two mirrors and a beam splitter, are the bare minimum requirements to conduct IFM. In practice, many more components are required to execute this scheme with electrons.

1.3.4 Components of a quantum electron microscope

Electron cavity: Analogous to the optical cavity required for high-efficiency IFM with photons, QEM requires an electron resonant cavity. A design for such a cavity is shown in Fig. 1.5. To take advantage of the existing electron gun column and the collimation optics, this cavity is designed to be integrated inside a conventional field-emission SEM. For better control over the convergence of the electron beam inside the cavity, the use of tetrode electron mirrors is proposed. Additionally, one of the two mirrors could be integrated with a diffraction grating (to be discussed shortly).

Gated mirrors: In this QEM design, in-coupling of an electron from the SEM source into the cavity is proposed to be achieved by a gated mirror: a slight lowering of the mirror potential allows the electron beam to enter the cavity and a quick return to the original voltage traps the in-coupled electron inside. This action must take place over a time scale of a few nanoseconds, on the same scale as the time for an electron round trip inside the cavity. After the desired number of round-trips is achieved, a similar mechanism is used to out-couple the electron towards the imaging optics.

Beam splitter and coupler: As was mentioned above, splitting and re-coupling of an electron is not as trivial a task as using partially silvered mirrors in the case of light. Using diffraction to split an incident electron beam into sample and reference beams is the method used in this design of QEM as shown in Fig. 1.5. These two beams, existing in a superposition state, play the role of beams in upper and lower branches of the Mach-Zehnder-interferometer model discussed in the previous section. Electron diffraction is often achieved using thin crystals: the periodic atomic structure of a thin crystal (e.g. silicon) can diffract a

(20)

transmitting beam. In fact, in a TEM, an objective lens recombines the diffracted beams onto an image plane to produce an image of the sample. However, the disadvantage of using thin crystals as beam splitters is the loss due to scattering or absorption of the electrons. This loss is especially an issue considering that for high-efficiency IFM, the electron is required to transmit though the beam splitter many times.

Another approach to electron diffraction is by integrating an electron mirror with a diffraction grating: diffractive electron mirror (DEM). This integration could be achieved by applying the mirror potential to a one-dimensional topographical grating instead of a flat electrode. Unlike thin crystals, a DEM is a phase-grating and does not scatter the incident electrons. Therefore, DEMs could be more appropriate for a large number of round-trips as required by QEM. DEMs will be the main focus of this thesis and will be discussed in details in the following chapters.

Sample: Analogous to the opaque object in the EV IFM scheme, the sample plays a crucial role in QEM by breaking the superposition state between the sample and the reference beams. An ideal sample for QEM would be one with completely opaque and fully transparent regions: black and white pixels, respectively. The sample holder could move the sample in two dimensions and bring each pixel into the path of the sample beam. An argument could be made that biological samples do not resemble this ideal sample; however, QEM could take advantage of the difference between the scattering efficiency of different regions of the sample to produce a silhouette of the sample. The gain here is the considerable reduction in beam-induced damage that is guaranteed by IFM.

Detector: After the desired number of electron round-trip has been achieved, the electron is out-coupled towards imaging optics and ultimately a detector to read out the results of IFM as shown in Fig. 1.5.

(21)

Electron

Gun

SEM

Optics

Gated

Mirror

Resonant

Cavity

Diffractive

Mirror

Sample

Reference

Beam

Sample

Beams

Readout

Optics

Opaque (black) Pixel

Transparent (white) Pixel

In-coupling

Out-coupling

Figure 1.5: Design for an interaction-free measurement scheme with electrons: quantum electron micro-scope (QEM). This design consists of an electron resonant cavity mounted inside a conventional field emis-sion SEM. A diffractive electron mirror, analogous to the beam splitter in the optical setup, splits/re-couples the electron beam. In the absence of an opaque object (white pixel), the electron probability density would have maxima corresponding to the sample beams. With an opaque sample present (black pixel), conse-quent measurement on the electron would indicate that the electron exhibited the reference beam inside the cavity. Figure courtesy of Marco Turchetti.

(22)

Chapter 2 Introduction to electrostatic electron

optics

Many different electron optical components are used in electron microscopy. In this chapter, two of these components relevant to quantum electron microscopy will be discussed: the electrostatic lens and the tetrode electron mirror. In addition, the focusing properties and non-idealities dictated by the electric field surrounding these components will be discussed.

In writing of this chapter, I frequently referred to Pieter Kruit’s very helpful lecture notes on charge particle optics [36].

2.1 Electrostatic electron lenses

In light optics, most lenses are made out of carefully shaped glass or other optically transparent materials with the appropriate refractive index. In the case of an ideal lens, light passing through the optical axis will not experience any deflection while the beams entering the lens at a point away from the center of the lens deflect by an angle linearly proportional to the distance between the point of entry and the optical axis. Thus, an ideal converging lens can focus a collimated beam onto a focal point.

One approach to achieving similar focusing behavior for an electron beam is the use of electrostatic lenses. This family of electron lenses focus a beam by applying an electrostatic force on the passing elec-trons. This force must be radially symmetric and have components towards the optical axis for a converging lens. Furthermore, for an ideal lens, this force must grow as the point of entry of the electron gets farther from the optical axis, closer to the edges. This requirement is paramount for focusing and ensures that an electron entering the lens near the edges experiences a larger deflection angle compared to one entering

(23)

the lens near the optical axis.

Two-aperture accelerating or decelerating lenses are simple cases where the appropriate field lines for focusing could be realized. As can be seen in the trajectory simulations in figure 2.1, under the influence of the the inwardly-directed radial components of the electric field near the edges of the apertures, an incoming collimated beam focuses into a spot on the other side of the lens. It is interesting to note that in the case of an accelerating lens (figure 2.1), the radial component of the electric field produces a force vector towards the optical axis near the first aperture, forcing the beam to converge. However, the force vector due to the radial component of the electric field near the second aperture point in the opposite direction, away from the optical axis. The subtlety here is that the magnitude of diverging force due to the fields near the second aperture is smaller that that of the converging force near the first aperture. This phenomenon could be justified in two ways. First, immediately after entering the lens, the beam converges and is consequently closer to the optical axis. And as it was discussed above, electrons near the optical axis experience a smaller amount of deflection. Second, the electrons experience acceleration in the lens and hence spend less time under the influence of the field lines near the second aperture. It could be shown with similar reasoning that the decelerating lens will also act as a converging (positive) lens.

Figure 2.1: Electron trajectory simulation of a collimated beam (blue lines) entering two-aperture lenses of (a) accelerating and (b) decelerating types. In both cases, the aperture electrode on the right is grounded while the one on the left is biased positively to 8000 V for the accelerating lens and biased negatively to −950V for the decelerating lens. Red arrows represent the direction of the electric field.

(24)

For many applications, it is desirable to preserve the potential energy of the electrons after focusing. Two-aperture lenses fail to do so. Three-aperture (einzel) lenses however preserve the potential energy of the electrons. Figure 2.2 shows an accelerating einzel lens focusing a collimated electron beam. Again, it is worth noting that neither accelerating nor decelerating einzel lens acts as a negative (diverging) lens. If fact, there are technically no diverging electron lenses. This is not to say that there are no field conditions under which an electron beam could diverge. In fact, a flat electron mirror (not a lens), which will be discussed shortly, could diverge an electron beam.

Figure 2.2: Simulation of an accelerating einzel lens. The incident collimated electron beam (blue rays) accelerated to 1 kV is focused on the right side of the lens operating at 5000 V (central aperture electrode). The side aperture electrodes are grounded. Thick red arrows represent the direction of electric field and the contours depict equipotential surfaces.

To change the focusing power of an electrostatic lens for a given electron acceleration voltage, one would typically increase the potential applied to the apertures to generate a larger electric field. In other words, the larger the electric field, the shorter the focal length, f, will be. The limit to achieving arbitrarily small f is electrical breakdown or arcing between the apertures of the lens at sufficiently high electric fields. In vacuum, over distances of the order of millimeters, an electric field of ∼ 10 kV/mm may be a higher limit one can achieve with relative ease without significant arcing. For higher electron energy applications such as transmission electron microscopy (TEM) where acceleration voltage routinely exceeds 100 kV, such a field entails very large voltages being applied to lenses. Therefore, impracticality is one of the reasons why magnetic lenses and not electrostatic lenses are used in high-energy electron microscopy. However, electrostatic lenses remain an inexpensive and relatively simple solution for lower-energy applications.

Spherical aberration is a type of geometrical aberration that is a character of a non-ideal lens which causes collimated incident beams to be focused at various focal lengths depending on the distance between the point of entry into the lens and the optical axis. In electron lenses, spherical aberration entails over-focusing of beams far away from the optical axis (marginal rays) and under-over-focusing of beams near the optical axis (paraxial rays). As a result, a collimated electron beam cannot be focused down to a single

(25)

point; rather a disc of focus (disc of least confusion) forms. The consequence of severe spherical aberration is limiting imaging resolution to the size of the disc of least confusion.

Although there are no easy or inexpensive ways to correct for spherical aberration, there are known techniques to mitigate its effects: (1) using accelerating lenses as opposed to decelerating ones, (2) reducing the focal length by going to higher field strengths, and (3) increasing the aperture diameter of the lens. In addition to these rules of thumb, the negative effects of spherical aberration could be largely avoided if the incident beam is sufficiently narrow and near the optical axis as opposed to the edges.

The spread of electron energy produced by a source entails that not all electrons will be focused at the same focal point. Faster electrons will focus farther away than slower electrons. This effect is known as chromatic aberration.

2.2 Electrostatic tetrode mirrors

The simplest way to reflect an electron is to bias a flat electrode to a negative potential higher in magnitude than that of the equivalent electron acceleration voltage. An electron incident on such an electrode will decelerate until it loses all its kinetic energy to potential energy at the classical turning point after which it will accelerate away from the electrode to until it reaches its original velocity. Another simple way to make an electron mirror is to charge up an insulator, e.g. teflon sphere, inside a scanning electron microscope (SEM) until it repels the consequent lower-energy electrons. This method, however, is less controlled and will not be discussed further.

The issue with the simple single electrode flat mirror introduced above is that the field lines are not terminated on a controlled surface. For instance, if such a mirror is placed inside an SEM, below the pole piece, field lines will form between the flat biased electrode and the grounded pole piece. This formation of field lines leads to unpredictable lensing effects which will diverge the reflected electron in an unpre-dictable way. To avoid this issue, it is important to place an aperture electrode above the flat electrode to terminate the field lines controllably. Figure 2.3 shows equipotential surfaces forming when a grounded aperture electrode, referred to as a cap, is placed above a negatively biased flat electrode. These equipo-tential surfaces and their corresponding field lines cause the reflected electrons to diverge away from the optical axis: negative lens effect.

(26)

Figure 2.3: Simulation of a flat mirror (m) capped with a grounded aperture electrode (C1). A collimated electron beam (blue lines) accelerated to 1 kV gets reflected by the mirror biased to −1010 V. The cap electrode is grounded. The thick red arrows represent the direction of electric field and the contours show equipotential surfaces.

2.3 Tetrode electron mirrors

A tetrode electron mirror is formed by integrating an electron mirror (mirror+cap) and an einzel lens. This integration gives the added benefit of controlling the convergence angle of the reflected beam. Tetrode electron mirrors are referred to by different terms in different applications. They are called immersion objective lens or cathode objective lens in emission microscopy [9]. The term ‘tetrode’ refers to the four electrodes used to construct a simple tetrode mirror. These electrodes are (1) mirror, (2) cap-1 or C1, lens (sometimes called C2), and finally cap-3 or C3 electrodes. Figure 2.4 demonstrates three simulated tetrode mirrors with various lens voltages to achieve the desired convergence angle for the reflected beam.

(27)

Figure 2.4: Simulation of a tetrode electron mirror operating at three different lens (L) voltages to control the convergence angle. The incoming rays are not depicted in this simulation; electrons are generated on the surface of the mirror. The mirror (m) electrode in all cases is set to −1010 V and both cap electrodes (C1 and C3) are grounded. The blue lines represent electron trajectories traveling from left to right. Colored lines represent equipotential surfaces. (a) VL= 2000V, (b) VL= 5000V, (c) VL = 1000V

(28)

Chapter 3 Diffractive electron mirror as a loss-less

electron beam splitter

As was discussed in Chapter 1, electron diffraction could be the means for coherent splitting and re-coupling of an electron beam to create the two states necessary for realization of interaction-free mea-surement in QEM. Diffraction could be thought of as a far-field consequence of electron interaction with a periodic structure which induces a disturbance in the wavefront of the electron. This disturbance could affect the electron wavefront in one of two ways: amplitude or phase. The former could be done by an am-plitude gratings, e.g. transmission grating, and the latter could be done by phase gratings, e.g. a diffractive electron mirror (DEM). In this chapter, transmission beam splitters will be briefly discussed, followed by a more in-depth discussion of DEMs and their use in QEM as lossless electron beam splitters.

3.1 Electron diffraction by means of a transmission grating

Periodic atomic structures in thin crystals have long been used to diffract electron beams in transmission. Recently, Agarwal et al. demonstrated the use of such thin crystals in the construction of a micrometer-sized Mach-Zehnder interferometer for IFM applications [37]. They propose that with some fabrication improvements, such thin-crystal beam splitters along with an opaque, sample could be used to realize the Elitzur and Vaidman IFM experiment with electrons. It is important to note that since their mode of operation is transmission, relatively high electron energies are required to ensure minimal inelastic scattering; in their work, Agarwal et al. used 200 kV electrons in a TEM.

Thin crystals are not the only periodic structures that could diffract electrons. As McMorran et al. experimentally demonstrated, an artificially fabricated diffraction grating could be used to split a

(29)

mitting electron beam [38]. They fabricated a free-standing periodic structure, a one-dimensional trans-mission grating, and used it to diffract an electron beam in an SEM. The schematic of their experiment is shown in Fig. 3.1a. Imaging a metal wire placed under the grating and observing repeated and laterally shifted copies of the same image on either side of the central bright image was used as proof of diffraction (Fig. 3.1b). This effect is due to focused higher order diffracted beams scanning the sample and producing secondary electrons, just as the primary beam does.

Objective lens Aperture Grating enclosure Diffraction grating Background SE sample (wire) Everhart-Thornley detector (a) (b) 20 μm Image column avg. grey scale val.

Figure 3.1: (a) Schematic of an experiment by Mcmorran et al., showing a transmission diffraction grating acting as an electron beam splitter in an SEM. (b) SEM micrograph resulted from the focused primary beam and the diffracted beams scanning a metallic wire and producing secondary electrons (SE). Figure adapted from [38]

3.2 Electron diffraction by means of a reflective grating

To realize truly lossless electron beam splitters, one cannot rely on electron transmission through matter. Even at high electron energies and for thin crystals, the probability of inelastic scattering could not be removed altogether. However, a DEM with a sinusoidal potential surface could impart the necessary phase shift onto the electron wavefront, the far-field consequence of which could be diffraction. Note that the electron has no direct interaction with the physical surface of the mirror; only potential surfaces. Hence, the efficiency of such a beam splitter could in theory be 100%. For simplicity, the two major components of a DEM could be treated separately. Namely, in this section, the quantum treatment of a flat electron mirror will be considered before incorporating the diffraction grating component.

(30)

3.2.1 Flat electron mirror

As was introduced in Chapter 2, an electron mirror in its simplest form consists of a flat electrode to which the mirror potential is applied and an aperture electrode (cap electrode) at which the electric field lines can terminate. Figure 3.2 shows an electron mirror with its various regions according to potential energy for a given electron energy E. Assuming that the cap electrode could confine the electric field in between the two electrodes, an electron on the left side of the mirror could be considered to be in a field-free space: region I. In its trajectory in the +x-direction (right), the electron begins to experience a linearly increasing potential energy in between the two electrodes: region II. As the potential energy grows, the electron slows down until at x = a, it loses all its kinetic energy to potential energy. This point is the classical turning point where the potential energy equals the initial energy of the electron. Classically, the electron will then begin accelerating in the −x-direction away from the mirror until it reaches its initial velocity in the opposite direction in region I.

Figure 3.2: Different regions of a flat electron mirror according to the potential energy: (I) field-free region x < 0where U(x) = 0; (II, III) retarding potential region 0 < x < b where U(x) increases linerarly with x; (IV) constant potential region: the physical surface of the mirror electrode. Classical tunning point occurs at x = a since U(x) = E where E is the electron energy in the field free region. Figure adapted from [39]

Since there are no time-dependent potentials in this problem, time-independent Schr¨odinger equation could be setup with various potential terms, U(x), to describe the electron in each region:

d2ψ dx2 +

2m

~2 (E − U (x)) ψ = 0 (3.1)

(31)

x-dependent potential energy. In region I, U(x) = 0 and hence Eq. 3.1 simplifies to d2ψI dx2 + 2mE ~2 ψ I_{= 0,} _(3.2)

which is the simplest form of the Schr¨odinger equation the solution to which consists of forward- and backward-propagating terms:

ψI(x) = C1exp (ikx) + C2exp (−ikx) (3.3)

where C1and C2are constants, determined by enforcing boundary conditions, and k = p2mE/~2. Note

that the forward- and backward-propagating terms in Eq. 3.3 represent the steady state superposition of the incident and reflected electron beams.

In regions II and III, the potential energy increases linearly with x. Assuming that the classical turning point of the electron, as shown in Fig. 3.2, is at x = a, i.e. U(x = a) = E, we can express the retarding potential in these regions as:

U (x) = E

ax (3.4)

hence the Schr¨odinger equation could be expressed as:

d2_ψII,III dx2 + k 2 a − x a ψII,III = 0. (3.5) Let ζ = (ka)2 3 x−a a

and Eq. 3.5 simplifies to [39]:

d2ψII,III dζ2 − ζψ

II,III _{= 0.} _(3.6)

The above equation is called the Airy equation with Airy function solutions:

ψII,III = C3Ai(ζ) + C4Bi(−ζ) (3.7)

known to describe the behavior of an electron inside a linear retarding potential and near classical turning points [39, 40, 41]. Figure 3.3 shows the behavior of these Airy functions. As can be seen, both functions have an oscillatory behavior when ζ < 0. However, beyond the turning point, ζ > 0, Bi(ζ) shows exponential growth which is unphysical. Therefore, we must set C4 = 0.

(32)

Figure 3.3: Behavior of Airy functions. Both Ai(ζ) and Bi(ζ) have oscillatory behavior for ζ < 0 with increasing wavelengths as they get near ζ = 0. Ai(ζ) exponentially decays when ζ > 0 and correctly captures the behavior of an electron for this region. Bi(ζ) cannot describe the behavior of an electron near and beyond the classical turning point since it exponentially increases (unphysical).

With the Bi(ζ) term eliminated, the solution to the Schr¨odinger equation in regions II and III will simply be

ψII,III = C3Ai(ζ) (3.8)

the behavior of which is shown in Fig. 3.3. Note that as the electron gets closer to the classical turning point, its wavelength increases but to a finite maximum length. This behavior is at odds with the classical theory according to which the electron wavelength blows up to infinity. It can also be seen from Fig. 3.4 that the amplitude of the electron probability density reaches its maximum just before the classical turning point suggesting that ζ = 0 (x = a) is the most likely place to find the electron.

Even though the electron wave function does not propagate into region III and quickly decays, the fact that the probability density extends into region III is another deviation from the classical theory which cannot explain quantum tunneling into energy barriers larger than the electron energy.

3.2.2 Corrugated electron mirror

Figure 3.5 shows the potential surfaces near a DEM. As can be seen, a DEM is very similar to a flat electron mirror for the most part. In fact, the only difference between the two is manifested very close to the surface of the mirror electrodes: flat potential surfaces form above a flat electrode while sinusoidal potential surfaces form above a corrugated electrode. This similarity allows us to use largely the same quantum treatment presented in the previous subsection with the exception that now, the potential depends on y (parallel to the plane of mirror) as well as on x (orthogonal to the plane of mirror). Hermans et al. treats

(33)

Figure 3.4: Probability density function of an electron in the retarding field above a flat mirror. The maxi-mum amplitude occurs near the classical turning point–the most likely place to find the electron

the y-dependence as a sinusoidal perturbation to the original problem and shows spreading of probability current density in the y-direction due to electron reflection off of a curved surface as can be seen in Fig. 3.6 [39].

Figure 3.5: Electrostatic simulation of equipotential surfaces above a diffractive electron mirror. Sinusoidal potential surfaces form only within a very small distance above the physical surface of the grating. The grating pitch and height are 500 nm. Simulation done on the commercially available software Lorentz 2E

The sinusoidal potential surfaces forming above the grating mirror are periodic along y and exponen-tially decaying in amplitude along x and could be expressed as:

U (x, y) = Exx − ExAx. exp −2πx d cos 2πy d (3.9)

(34)

max-Figure 3.6: Addition of a sinusoidal perturbation to the quantum model of a flat mirror leads to spreading of the probability density (arrows on the left) in the y-direction which indicates diffraction. Figure from [39]

imum amplitude of the sinusoidal potential just above the surface of the diffractive mirror, and d is the period (pitch) of the sinusoidal potential surface. Note that x is the direction of propagation and y is par-allel to the surface of the mirror. In this case, the variation in potential surface is taken to be along y with no variation along z: one-dimensional grating.

When a collimated electron beam is incident on such a sinusoidal potential surface, different parts of the beam encounter different phase shifts upon reflection. In far-field, this modulation of phase shift leads to constructive and destructive interferences. Where different parts of the beam interfere constructively, diffraction spots appear in accordance with certain diffraction angles. As can be seen from Eq. 3.9, the amplitude of the sinusoidal potential decays exponentially. Therefore, not far from the physical surface of the grating, potential surfaces flatten to resemble those of a flat mirror. This suggests that for the electron beam to encounter the diffracting effects of the sinusoidal potential, the classical turning point must occur very close to the physical surface of the diffractive mirror. The rule of thumb is for this distance to be close to the pitch of the grating (0.1 − 1 µm).

3.2.3 Diffraction angle and the relevant electron energy

As it was discussed earlier, the phase shift modulation imparted on the electron wavefront after reflection off of a sinusoidal potential surface leads to diffraction spots in the far-field. The angles (with respect to normal) at which these spots appear depend on the de Broglie wavelength of the incident electron as well as the pitch of the grating:

sin θ = λ0

(35)

where d is the pitch of the grating and λ0is the de Broglie wavelength of the electron in free space, before

entering the retarding potential. Generally, larger diffraction spot separation (large θ) is of interest for more convenient observation of such spots. Using diffraction gratings of smaller pitch and lower electron energies would increase θ. However, there are practical limitations preventing us from arbitrary pursuit of both these factors which will be discussed in the next section.

As shown in Fig. 3.4, the wavelength of the electron increases as it approaches the classical turning point due to losing kinetic energy in the retarding field. Looking at Eq. 3.10 and assuming diffraction occurs near the classical turning point where the electron wavelength has increased, one might then wonder why the diffraction angle is affected by the free space wavelength λ0and not by the elongated wavelength near

the point of reflection/diffraction. Although the immediate diffraction angle inside the field region does depend on the elongated wavelength of the electron, after reflection, the electron takes a curved trajectory while accelerating only in the direction of propagation and emerges out of the field region at an angle determined by Eq. 3.10. In other words, the apparent diffraction angle observed in the field-free region is dependent solely on the initial electron wavelength λ0and the grating pitch d.

3.3 Design of a proof-of-principle experiment

Figure 3.7 shows the design for a proof-of-principle experiment to demonstrate the beam splitter function of a DEM inside a conventional SEM. A sample, e.g. a micrometer-sized silicon cantilever atop a 5-axis nano-positioner is positioned near the optical axis of the SEM objective lens. The nano-positioner is itself mounted on a 3-axis SEM stage (x,y and z). Below the sample, also on the 3-axis stage, a tetrode electron mirror is positioned such that its optical axis is aligned to that of the SEM objective lens.

As a focused beam scans the top surface of the sample, secondary electrons are produced, which when collected by the positively biased in-lens detector of the SEM, an image of the sample could be formed. However, during the same scan period, some electrons get past the sample and continue their downward trajectory towards the tetrode mirror. The tetrode mirror could then reflect and refocus the electron beam towards the bottom surface of the sample, producing an image of the bottom surface in the same image frame. When the flat mirror is replaced by a diffractive electron mirror (Fig. 3.7b), the beam incident on the mirror will be both reflected and diffracted, leading to multiple focused beams on the bottom surface of the sample. Observation of multiple images of the sample in the same SEM micrograph could be evidence for achieving electron diffraction in reflection mode and hence a lossless electron beam splitter.

(36)

Figure 3.7: Design of a proof-of-principle experiment that could demonstrate a diffractive electron mirror functioning as lossless electron beam splitter. A focused electron beam scans the top surface of a sample placed near the optical axis, producing an image of the top surface of the sample. During the rest of the scan, some electrons get past the sample and continue their downward trajectory towards the electron mirror. (a) The flat tetrode mirror reflects and refocuses the beam towards the bottom surface of the sample, producing an image of the bottom surface. (b) The diffractive electron mirror reflects the diffracted beams towards the bottom surface of the sample. Multiple focused beams produce multiple images with a slight lateral shift corresponding to diffraction spot separation. In both cases the sample is placed on the back-focal plane of the tetrode lens. The sample is mounted on a 5-axis nano-positioner (not shown in this schematic). Electron detection is done using the in-lens detector positioned inside the electron column.

lens of the tetrode mirror. In practice, this is achieved not by moving the sample but by moving the back-focal plane by adjusting the voltage applied to the lens electrode. Placing the sample on the back-back-focal plane of the tetrode mirror ensures two requirements: (1) the incident beam focused on the sample plane will be collimated after entering the tetrode mirror, (2) the diffracted beams will be focused on the sample plane (Fourier plane). Less crucial but advantageous is adjusting the tetrode mirror voltages in concert with the SEM objective lens current such that the pivot point1_{of the SEM is imaged on the mirror surface.}

satisfying this condition ensures that as the beam scans, it will remain stationary at the plane of the mirror. Failure to satisfy this condition may cause the beam to enter the tetrode mirror system close to the edges where spherical aberration is exacerbated.

For a given electron energy, the magnitude of the negative bias applied to the grating mirror electrode dictates whether the mirror acts as a diffractive mirror or a flat mirror. The former occurs at biases only

(37)

very slightly above the electron energy such that the classical turning point lies on a sinusoidal potential surface while the latter occurs when the negative bias is large enough such that the electron reflects off of a flat potential surface above the grating mirror. This design provides flexibility in switching between the two types of mirrors.

3.3.1 Tetrode electron mirror voltages

To satisfy the aforementioned requirements of this experiment, voltages applied to different electrodes of the tetrode mirror must be fine-tuned. To get an estimate of these values, two-dimensional electron trajec-tory simulations were performed on a commercial software (Lorentz-2E; Integrated Engineering Software). A flat tetrode mirror was modeled using the electrostatic boundary element simulation followed by an elec-tron trajectory simulation where the elecelec-trons start near the surface of the mirror with no kinetic energy. This resembles the classical turning point where the electron loses all its kinetic energy. Figure 3.8 shows the results of this simulation. By including several electrons with the same initial angle and separated in the y-direction, we can simulate a collimated electron beam above the mirror. As can be seen from Fig. 3.8, as the electrons accelerate away from the mirror surface, under the influence of electric field lines near cap-1 electrode, the beam initially begins to diverge. However, with the appropriate voltage applied to the lens electrode (cap-2), the beam then converges to a focus on the sample plane.

We can treat the voltages obtained in the above simulation as a starting point for an electron round-trip trajectory simulation. Figure 3.9 shows the trajectory of 3 keV electrons initiated at the back-focal plane of the lens (x = 12 mm), traveling in the −x-direction. The initial divergence angle is set by the outermost rays to be 10 mrad. As can be seem, the electrons become collimated above the mirror due to the fields surrounding the tetrode mirror. After reflection, the electrons complete a round-trip ending up approximately where they started at x ∼ 12. It is interesting to note that for the inner rays (small divergence angle), due to accurate overlap, the incident and the reflected beams are not distinguishable. It is only for the outermost rays that we can distinguish the two rays. This result is due to the more severe effects of spherical aberration on the beams that enter the tetrode lens near its edges.

It is important to note that some level of voltage discrepancy is to be expected between electron tra-jectory simulations and experiments. Thus, the above simulations serve as a useful starting point for the voltage range that should be applied during the experiment.

(38)

-0.025 -0.015 -0.005 0.005 0.015 0.025 0 5 10 15 20 y (m m ) x (mm) 1mm 1.3 mm 1.3 mm Vm Vc1 VL1 Vc3 sample optical axis

Figure 3.8: Two-dimensional trajectory simulation of a collimated electron beam. Top: electrons start at the classical turning point with no energy (left), accelerate away from the mirror and get focused on the back-focal plane of the tetrode lens (dashed line). Beam diameter was taken to be 20 µm (simulation symmetric about y = 0). By sweeping the voltages applied to the tetrode mirror, the desired back-focal plane was positioned at x = 12 mm (sample plane). Bottom: tetrode mirror electrodes arrangement: Vm = −3020

V; Vc1 = 1493V; VL = −3860V; Vc3 = 0V. Simulation done on the commercially available software

Lorentz 2E.

3.3.2 Electron energy

As was mentioned in Chapter 1, due to diffraction limit, high electron energies are required for atomic resolution imaging of biological samples. Even for imaging of the silicon cantilever in the context of the experiment introduced above, higher energy electrons are less susceptible to stray fields in the chamber and could produce higher-resolution images. However, for the sake of this proof-of-principle experiment the chosen range of electron energy was 1−3 keV which was readily attainable in our field emission SEM. Two major reasons lead to this choice of energy range.

First, the mirror at the bottom of this setup must be biased to a negative potential equivalent to the energy of the incident electron. For instance, for a 3 keV electron beam, the potential applied to the mirror must be larger in magnitude than −3000 V. As a result for significantly higher energies, it could become

(39)

-0.12 -0.07 -0.02 0.03 0.08 0 5 10 15 20 y (m m ) x (mm)

Figure 3.9: Trajectory simulation of electrons completing a round trip after getting reflected by the flat tetrode mirror. Starting point: sample plane (x = 12 mm). Using the exact same voltage settings as shown in Fig. 3.8, 3 keV electrons with various divergence angles propagate towards the +x-direction. Since the initial beam is focused on the back-focal plane of the tetrode lens (x = 12 mm), the beam becomes collimated above the mirror (beam diameter ∼ 90 µm). Under the influence of spherical aberration, the outermost beam (purple) does not take the exact same trajectory after reflection and hence the incident and the reflected beams are distinguishable (imperfect overlap). This effect is minimal for the inner beams due to their proximity to the optical axis. Simulation done on Lorentz 2E.

inconvenient to supply the appropriate voltage to the mirror. These inconveniences stem from the need for voltage-grade cables and electrodes as well as the safety risks and high costs associate with high-power voltage supplies.

Second, according to Eq. 3.10, under small angle approximation, diffraction angle is directly propor-tional to the field-free wavelength of the incident electron. This entails that the diffraction angle is in-versely proportional to the square root of energy: θ ∝_√1

E. Therefore, we can see that for a given grating

pitch, higher electron energies lead to a smaller diffraction angle. For a given diffractive-mirror-to-sample distance, smaller diffraction angles lead to smaller diffraction spot separations which in the case of this proof-of-principle experiment could make it challenging to observe distinct multiple images.

3.3.3 Grating dimensions

As can be seen from Eq. 3.10, for a given electron energy, DEMs with smaller grating pitches lead to diffraction at larger angels. This is obviously desirable in this experiment. However, as was mentioned earlier in this chapter, the smaller the grating pitch, the closer the incident electron has to get to the surface of the physical grating to bear the appropriate phase shift modulation. In fact, as a general rule of

(40)

thumb, the distance between the physical grating and the classical turning point should be roughly equal to the grating pitch. For very small grating pitches (< 100 nm), this puts the electron at risk of inelastic scattering. To avoid this issue, for this experiment, a pitch of 500 nm was chosen. A 3 keV electron beam incident on such a DEM would diffract at a diffraction angle of 44 micro-radians (µrad). Assuming that the the sample plane is 12 mm above the mirror, diffraction spots separation of the order of 500 nm could be expected.

Although simulation software Lorentz-2E does not take the phase of electrons into account and hence cannot simulate diffraction, y-directed “momentum kicks” could be incorporated in the trajectory sim-ulations to resemble diffraction. The magnitude of these momentum kicks could be approximated from |pyn| =

nh

d, where h is the Planck’s constant, d is the pitch of the grating and n is the diffraction order.

The angle of these momenta could be found from diffraction angles given by Eq. 3.10 for various orders. Figure 3.10 shows simulation of electrons starting their trajectories on the classical turning point of the flat mirror with no initial horizontal momentum and vertical momenta in directions associated with the zeroth diffraction order, θ0, first diffraction order, θ1 and second diffraction order θ2. These angles were

calculated for incident electron energy of 3 keV and grating pitch of 500 nm.

Figure 3.10: Mimicking diffraction by forcing the appropriate diffraction angles upon electrons starting from the left. Diffraction angles were calculated using Eq. 3.10 for a 500 nm pitch grating and a 3 keV elec-tron beam. The simulation settings are identical to those in Fig. 3.8 except for the initial angles. Diffraction spots form on the back-focal plane of the tetrode lens (inset): blue: zeroth order; green: first order; red: second order diffracted beams. Each starting point on the mirror is imaged on the image plane at x ∼ 25 mm. Simulation done on Lorentz 2E.

Diffractive electron mirror for use in quantum electron microscopy

Acknowledgments

List of Figures

Contents

Chapter 1

Avoiding radiation damage in electron

microscopy

1.1

Low-energy schemes

1.2

Low-dose schemes

1.3

Quantum mechanical solutions

(a)

(b)

Electron

Gun

SEM

Optics

Gated

Mirror

Resonant

Cavity

Diffractive

Mirror

Sample

Reference

Beam

Sample

Beams

Readout

Optics

Opaque (black) Pixel

Transparent (white) Pixel

In-coupling

Out-coupling

Chapter 2

Introduction to electrostatic electron

optics

2.1

Electrostatic electron lenses

2.2

Electrostatic tetrode mirrors

2.3

Tetrode electron mirrors

Chapter 3

Diffractive electron mirror as a loss-less

electron beam splitter

3.1

Electron diffraction by means of a transmission grating

3.2

Electron diffraction by means of a reflective grating

3.3

Design of a proof-of-principle experiment