Electron holography for fields in solids: Problems and progress

Ultramicroscopy 134 (2013) 126–134

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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic

Electron holography for fields in solids: Problems and progress$ Hannes Lichte n, Felix Börrnert, Andreas Lenk, Axel Lubk, Falk Röder, Jan Sickmann, Sebastian Sturm, Karin Vogel, Daniel Wolf Triebenberg Laboratory, Institute of Structure Physics, Technische Universität Dresden, Zum Triebenberg 50, 01328 Dresden, Germany

art ic l e i nf o

a b s t r a c t

Available online 11 June 2013

Electron holography initially was invented by Dennis Gabor for solving the problems raised by the aberrations of electron lenses in Transmission Electron Microscopy. Nowadays, after hardware correction of aberrations allows true atomic resolution of the structure, for comprehensive understanding of solids, determination of electric and magnetic nanofields is the most challenging task. Since fields are phase objects in the TEM, electron holography is the unrivaled method of choice. After more than 40 years of experimental realization and steady improvement, holography is increasingly contributing to these highly sophisticated and essential questions in materials science, as well to the understanding of electron waves and their interaction with matter. & 2013 Elsevier B.V. All rights reserved.

Keywords: Electron holography Phase contrast Structures and fields in materials Nanofields in solids

perfect imaging, the image wave results

1. Introduction Abbe introduced wave optics in microscopy. This implies that objects may modify both amplitude a and phase φ of the traversing wave. Consider a wave leaving an object in weak object approximation obj ¼ a expðiφÞ≈1−t þ iφ

ð1Þ

with object transmission t ¼ 1−a. In the linear approximation, the imaginary unit i indicates that φ is the phase of the wave. Under

☆ Dedication: Owen Saxton pioneered Numerical Image Processing and built up Semper-Software. Thanks to his efforts, as guests from Tübingen in the MPI for Biochemistry, Martinsried, in 1985 we had the chance to reconstruct with Semper the numerical waves from our first atomic resolution holograms and find the way for a-posteriori aberration correction. Moreover, Owen developed the GerchbergSaxton algorithm for determining the electron wave from diffraction patterns and contributed to aberration correction. David Smith: Since 1987 we have steady exchange in cutting edge problems in materials science and discuss possible applications of holography with David and Molly: The joint scientific interests developing in friendship across the ocean foster questions and solutions on both sides such that we learned a lot on materials issues. They belong to the worldwide experts in the field of holography. Dirk Van Dyck: is always interested in the basics. So we discussed basic issues such as electron waves and coherence. We had many walks in beautiful places such as Tempe, Brügge, Giens, and Maui discussing these mysteries, in particular inelastic interaction and waves. Dirk developed focal series holography, and in joint endeavors like BRITE/EURAM and the Francqui Foundations he set up a frame for a fruitful relation with Antwerp. n Corresponding author. Tel.: +49 35121508910. E-mail addresses: [email protected], [email protected] (H. Lichte).

0304-3991/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2013.05.014

ima ¼ 1−t þ iφ:

ð2Þ

Alas, the detectable intensity distribution Int ¼ ima iman ¼ 1−2t

ð3Þ

(neglecting the quadratic terms) does not show any phase information, i.e. no phase contrast. Zernike understood that the phase is not showing up because it is weighted by the imaginary unit i. Therefore he proposed a plate shifting the phase in Fourier space by π=2 in order to swap i ¼ expðiπ=2Þ from the phase φ to the transmission t; then in real space the image wave ima ¼ 1−it−φ

ð4Þ

results. Please note that in this image wave t modulates the phase, and φ takes the role of transmission. Consequently, the intensity Int ¼ 1−2φ

ð5Þ

displays the object phase in so-called phase-contrast and the transmission disappears [1]. Phase contrast in light optical microscopy meant a huge benefit for biology where most interesting objects are phase objects; therefore, in 1953 Zernike was awarded the Nobel Prize in Physics. Most of the highly merited pioneers in electron microscopy at the beginning were not aware about the role of wave optics in TEM and not about phase objects. At a get-together, Ruska talked about the early TEM-history with the gist: at the beginning we thought the image contrast stems from inelastic interaction, and then we learned that it stems from elastic amplitude modulation, and finally we know that it is predominantly phase contrast.

H. Lichte et al. / Ultramicroscopy 134 (2013) 126–134

“The readers need not fear that we are talking much about electron waves….” [2] In those days, experimentalists were mostly not acquainted with electron waves. However, theoreticians knew better. As early as 1933, Glaser introduced the index of refraction for electrons [3] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! k ðU acc þ Vð r ÞÞn e !! !! nð r ; s Þ ¼ ¼ ð A s Þ: ð6Þ − n k0 hk0 U acc U acc is the accelerating voltage, and k and k0 are the wavenumbers in an electromagnetic field and in field-free space, respectively. Therefore, a wave propagating through an area with ! ! the electric potential Vð r Þ and a magnetic vector-potential A ! ! ! (with induction B ¼ curl A ) along the path s collects a phase R ! φ^ ¼ 2πk0 n ds. This is still gauge dependent in V and A . With respect to field-free space a gauge-independent phase difference R φ ¼ 2πk0 ðn−1Þ ds results, which, for the usual V⪡U acc , can be written as Z Z e Az ðx; y; zÞ dz ð7Þ φðx; yÞ ¼ s Vðx; y; zÞ dz− ℏ integrated along z-direction; the interaction constant is s ¼ e=ðℏvÞ with v the electron velocity. Consequently, electric and magnetic fields are essentially phase objects. This “Phase Grating Approximation” is valid for sufficiently weak fields mostly occurring on the micro- and nanoscale. Scherzer in 1949 pointed at the fact that also atoms are mainly phase objects. Since there was no way to realize the Zernike phase plate for phase contrast in a TEM, Scherzer showed that aberrations, in particular spherical aberration in combination with the “Scherzer focus”, in the back focal plane produce a phase distribution expð−iχðqÞÞ with “wave aberration” χðqÞ approaching a phase shift of π=2 over a wide band of spatial frequencies (“Scherzer band”), and hence mimic a Zernike phase-plate [4]. For describing the information transfer, he derived the Phase Contrast Transfer Function PCTF ¼ sin ðχðqÞÞ and the Amplitude Contrast Transfer Function ACTF¼cos(χ(q)). The Scherzer band of optimum phase transfer reaches from the upper limit (“point resolution”) qmax to the lower limit qmin ≈qmax =5; large-area phase structures larger than 1=qmin such as extended electromagnetic fields are not visible. Evidently, a perfectly aberration-corrected TEM does not reveal any phase contrast because of χðqÞ≡0, and hence PCTF≡0 and ACTF≡1; furthermore, at conventional imaging, the phase channel is blocked, because phases are never directly visible in a detector: an aberration corrected TEM is phase-blind. Recent efforts for introducing a Zernike phase plate in TEM are reported, e.g., in [5,6].

2. Electron interferometry Already in the 19th century, light interferometers had been built and very successfully used for phase measurements. Therefore it was very attractive to transfer this principle to electrons. The main problem was to find a high-performing beam splitter. Möllenstedt and Düker developed the electron biprism as a wave front beam splitter for off-axis electron interferometry in 1954 [7]. It is the analog of the light optical Fresnel biprism splitting a wavefront in two coherent partial waves: one wave goes though the object; the other one passes by through empty space resulting in the object exit wave aðx; yÞexpðiφðx; yÞÞ. At some distance downstream, e.g. by propagation or lens aberrations, the object exit wave builds up the image wave Aðx; yÞexpðiϕðx; yÞÞ, which is superimposed with the empty reference wave of modulus 1 at

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an angle β producing a cosinoidal interference pattern Iðx; yÞ ¼ 1 þ Aðx; yÞ2 þ 2jμjAðx; yÞ cos ð2πqc x þ ϕðx; yÞÞ: ð8Þ The carrier frequency is given by qc ¼ k0 β with the wave number k0 ¼ 1=λ. The interference term is weighted by the degree of coherence 0 ≤jμj ≤1 between the superimposed waves. The deviation Δsðx; yÞ of the interference fringes from a straight line gives the local phase distribution ϕðx; yÞ, which can easily be determined by the relation ϕ ¼ 2πqc Δs; this evaluation is simply done by means of a ruler. Möllenstedt and coworkers developed different paths of rays for electron interferometry and interference microscopy. They also performed abundant measurements of electric potentials in solids [8–12], e.g. the Mean Inner Potential and contact potential between different metals, as well as magnetic fields in and around microscopic objects, including the Ehrenberg–Siday–Aharonov–Bohm (ESAB)-effect [13], which was later confirmed by Tonomura [14]. For interpretation of the findings in terms of object properties, together with Lenz [15], they described the theoretical principles of electron interferometry and, applying Eq. (7) for both waves, they set up the basic relation for the resulting phase shift distribution ϕðx; yÞ ¼

e e V ðx; yÞ− Φðx; yÞ ℏv proj ℏ

ð9Þ

with electron velocity v, the “projected electric potential” R V proj ðx; yÞ ¼ Vðx; y; zÞ dz integrated through the object area, and ! ! the magnetic flux Φðx; yÞ ¼ ∮ A ðx; y; zÞ d s embraced by the two superimposed beams; because of comparison with the reference wave, the phase is gauge invariant. The ESAB-Effect [16,17] says that the magnetic phase shift occurs, even if the embraced magnetic field is distributed such that it vanishes at the trajectories of the electrons, and hence the electrons do not experience any classical Lorentz force. Based on the ESAB-effect, Flux Quantization in superconducting hollow cylinders was measured [18,19]. The biprism is a wave front splitter, like the double slit, in that different parts of a wave front are superimposed downstream. There were also very early experiments on electron interferometry [20] using a crystal (diffraction grating) as an amplitude splitter, which can be regarded as an analog of the Michelson interferometer using semi-transparent mirrors to split a wave into two coherent twin-waves; in each point of the wave front, twinwaves issue in different directions subsequently superimposed by optical means such as amplitude or wavefront splitters, or lenses. The latter occurs under usual TEM-imaging of a crystal. The amplitude splitter is superior as to spatial coherence requirements, if the twin-waves are again exactly brought to interference point by point; for example, a Michelson interferometer can be operated with a candle light. However, the splitter, i.e. in electron optics a diffraction grating or crystal must be perfect to avoid amplitude or phase modulation by defects, thickness variations and surface contamination. Also, the diffraction efficiency, depending on crystal tilt, is often insufficient, and usually more than two diffracted waves are issued, which are lost for the purpose of two-beam interference. Nevertheless, amplitude splitters became recently very interesting for the generation of special electron waves, such as vortex beams [21].

3. Electron holography Gabor understood that interferometry and diffraction are two sides of the same coin: one of the waves diffracted at an interferogram (Eq. (8)), considered as a diffraction grating, is a replica of the complete image wave [22]. Instead of evaluating an interferogram simply by means of a ruler, holography widens the evaluation in that it extracts from the reconstructed wave two images, i.e. the amplitude image Aðx; yÞ given by the local fringe contrast and the phase image

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ϕðx; yÞ representing the local phase shift. Cowley listed more than 20 ways of electron holography [23], but image plane off-axis holography and the corresponding reconstruction introduced in electron holography by Wahl [24] is the most powerful up to date. The evaluation of the hologram data, initially performed by diffraction of a LASER-beam at the hologram, is nowadays done by numerical image processing emulating the diffraction step in Gabor´s scheme by a Fourier transform and masking out the wanted Fourier spectrum of the image wave A expðiϕÞ. The lateral resolution obtained in real space after inverse Fourier transform, given by the opening of the mask according to Abbe's resolution formula, cannot exceed the triple fringe spacing in that at least 3 fringes cover one reconstructed pixel; in case of pure phase objects, one fringe is sufficient. Step by step, the basics of electron holography have been improved and an increasing number of applications has been shown, at the early stage mainly in the groups in Tübingen, Tokyo, Braunschweig and Bologna. Nowadays, many groups around the world are contributing to the furtherance and application to specific problems. Details can be found for example in [25–29]. 3.1. Figure of merit It became experimentally evident that the quality criteria of the reconstructed wave mainly are: field of view covered with fringes, fringe spacing giving lateral resolution, and fringe contrast determining the discernibility of small phase differences well above noise of the reconstructed wave. In a succinct way, the performance of electron holography can be assessed by a figure of merit, which concatenates field of view, lateral resolution and signal resolution [30]. The figure of merit is given by the information content of the reconstructed wave defined as Inf oCont ¼ nrec nφ

ð10Þ

with nrec the number of reconstructed pixels across the width w of the wave; by nrec ¼ 2qres w, it links field of view w with lateral resolution qres . Since one reconstructed pixel covers about 12 camera pixels in one direction (3 fringes sampled with 4 camera pixels each, see e.g. [25]), the field of view comprises nrec ≈180 reconstructed pixels with a 2kn2k camera. nφ is the number of phase steps distinguishable in phase range of 2π; the phase detection limit follows as δφlim ¼ 2π=nφ . It turns out that Inf oCont ¼

2π V inel snr NoiseFigure

ð11Þ

is given by contrast damping due to inelastic interaction V inel with the object as discussed later, and by the intended signal/noise ratio snr. The most essential part

NoiseFigure ¼

pffiffiffiffiffiffi 2π qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jμsc jV inst V MTF −lnðjμsc jÞ B2 ετDQ E

ð12Þ

ek0

describes the quality of the electron microscope by means of the degree of spatial coherence μsc of illumination, V inst contrast damping by instabilities, V MTF and DQ E, i.e. Modulation Transfer Function MTF and Detective Quantum Efficiency DQE of the camera, B brightness of electron source, k0 wave number, e electron charge, ε ellipticity of illumination, and τ exposure time. Usually, temporal coherence is by far sufficient hence need not be considered here. To give an example, for our FEI Tecnai F20 Cs-corr TEM operated in the Triebenberg-Lab, Inf oCont≈18; 000 can be reached; this means that a wave reconstructed with nrec ¼ 180 pixels allows measuring phase differences down to 2π=100 between adjacent pixels. This would allow distinguishing single light atoms such as carbon or oxygen with

2π=60…2π=50, recognizing single electron charges with 2π=30… 2π=100, however, by far not single Bohr Magnetons with 2π=105 . 3.2. Role of the detector The above MTF and DQE used for describing the performance of a camera give only a rough characterization of the influence of the camera on the signal finally obtained. In fact, for a comprehensive understanding one has to analyze the camera in more detail. More detailed investigations show that the above formula is valid for a specific type of CCD–cameras only. A general treatment of the noise transfer properties through the detection process is given by the noise spread function introduced by [31,32]. Furthermore, it is worth mentioning that the above noise figure only discusses the variances in the reconstructed phase image. For the determination of error bars after various image processing steps, however, it is necessary to consider the covariances also. They have to be calculated from the noise spread function and noise propagation through the reconstruction process. Incorporation of these details will allow a more precise understanding of the information content as to the role of the camera. 3.3. Optimized recording NoiseFigure must be made as small as possible. A minimum is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi found by maximizing jμsc j −lnðjμsc jÞ at jμsc j ¼ 0:61 selectable by means of the condensor optics. Furthermore, the contrast damping by instabilities must be minimized by a very careful installation in an appropriate, disturbance-free laboratory. Because of  U acc , the 2 reduced brightness B=k  B=U acc is, independently from the accelerating voltage U acc , given by the emission process; therefore field emission or, at least, Schottky emission has to be employed and optimized with respect to brightness B [33]. A high p potential ffiffiffiffiffiffiffiffiffiffi for improvement still rests in the camera, because V MTF DQ E is still substantially worse than an ideal detector. The ellipticity ε of illumination helps enhancing the current density for a given coherence perpendicular to the biprism direction; therefore, at first glance, the larger the better. However, it is restricted in that the point spread function resulting from lens aberrations and defocus has to be sufficiently coherently illuminated also in the direction parallel to the biprism filament. The curvature of the wave front resulting at elliptic illumination has to be considered at holographic correction of aberrations [34] only in a not aberration-corrected TEM. The only parameter freely available is exposure time τ. It should be as large as possible to collect the highest possible electron number per reconstructed pixel, of course, avoiding saturation of the camera. However, drift of biprism, object or currents of lenses and deflectors may reduce the fringe contrast V inst with increasing pffiffiffi τ; the guideline for optimizing is to maximize V inst τ. In our lab, exposure times up to 30 s are applied at medium resolution, at atomic resolution sometimes 20 s lead to satisfactory results. Cooper et al. reported about improving considerably the quality of medium resolution holograms at an exposure time of more than 2 min using a highly stable FEI Titan TEM [35]. A more determined method consists in recording a series of identical holograms, each at an exposure time preserving the contrast V inst , selecting the ones with best fringe contrast, reconstructing the waves, equalizing the phases as to offset and tilt, and finally cross-correlating the waves and averaging them. Following this procedure with N selected holograms, one can expectedly reduce NoiseFigure by a pffiffiffiffiffi factor N [36,37]. 3.4. Time averaging Irrespective of the applied method, we collect many electrons during exposure time. These electrons cannot be considered

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mutually coherent, since they are not in the same state; only the two superimposed partial waves, i.e. object and reference wave, of each electron are coherent with each other. Consequently, each electron produces its own hologram; however, visible is only one impact, and hence we have to collect an ensemble of many electrons, which resemble each other as much as possible (“ensemble coherence”). Of course, each single hologram reflects the specifics of the respective electron, such as energy and direction of illumination, according interaction with the object, respective object state, and effect of lens aberrations, etc. Mapping the electrons by their time t i of impact on the detector, we find the respective waves as Aðx; y; t i Þexpðiϕðx; y; t i ÞÞ possibly also depending on time. The final hologram displayed by many electrons is an incoherent average over the holograms of all single electrons. Under the reasonable assumption that the resulting degree of coherence μ of the collected electron ensemble does not depend on time, the interference term sums up as 2jμj τ ∑ Aðx; y; t i Þ cos ð2πqc x þ ϕðx; y; t i ÞÞ τ ti ¼ 0

ð13Þ

over exposure time τ. The damping by the degree of coherence reduces the signal/noise ratio discussed subsequently. The reconstructed wave follows as imarec ðx; yÞ ¼

jμj τ ∑ Aðx; y; tÞexpðiϕðx; y; t i ÞÞ: τ ti ¼ 0

ð14Þ

Surprisingly, in off-axis electron holography the reconstructed wave represents the coherently averaged waves of all elastically scattered electrons, which has never existed in the TEM, because the electrons never have been in one wave. If the object changes during exposure time, e.g. under irradiation, the reconstructed wave represents object properties coherently time-averaged. For example, in the case of an atom considered as a pure phase object that hops out of the field of view after half the exposure time, an amplitude modulation will occur and the phase modulation will be halved. Therefore, the intensity of the averaged wave will not agree with the incoherently averaged intensity of the same electrons in conventional imaging; in the above example with the hopping atom as pure phase object, the conventional intensity would show no contrast at all. This makes a significant difference compared to the conventional imaging process, which has to be considered at interpretation [38]. 3.4.1. Time resolution For the investigation of time-dependent phenomena, a short exposure time τ is desirable. In Eqs. (11) and (12) exposure time pffiffiffi shows up in the NoiseFigure and hence Inf oCont∝ τ. This means that, keeping the number of reconstructed pixels constant, reducpffiffiffi tion of exposure time would worsen the signal resolution by τ. For example, at τ ¼ 1 ms, compared to 10 s, the phase detection limit would be a 100 times worse, close to 2π, meaning that only very strong phase structures would be detectable from a single hologram. Reasonable time resolution will only be reachable, if brightness B and Detection Quantum Efficiency DQ E in Eq. (12) can pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be improved such that B τ DQ E remains large enough. 3.5. Inelastic interaction The phase signal in the reconstructed wave stems from elastic interaction meaning that an electron has exactly the same energy after leaving the specimen as it had before interaction. However, a significant fraction of interaction is inelastic with energy transfer to and from the object. Inelastic interaction deposits energy in the object. By this, the object may change its structure and consistency, in particular at the most interesting real structures such as defects, interfaces and surfaces; furthermore, it may charge under

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the beam e.g. by emission of secondary electrons, and provoke currents percolating through the specimen hence producing local potential changes. This severely hampers the interpretation of findings in terms of the object. Of course, by these inelastic events, a beam electron loses energy and hence its wave optical properties change. The change of wavenumber is negligibly small. However, a huge effect arises in that at interference with an elastically scattered or unscattered electron wave the energy loss ΔE produces a beat frequency Δν ¼ ΔE=h with h ¼ 4:135  10−15 eV s, i.e. the interference fringes positions fluctuate [39]. With the usual exposure times in the range of seconds and more, the interference pattern is wiped out at an energy transfer of more than ΔE≈10−15 eV, and hence, depending on exposure time, coherence between the two waves disappears. In any case, an electron that has suffered a larger energy transfer is incoherent with the elastic and the unscattered ones hence contributes only to the background of the hologram. This was the reason for assuming that the wave reconstructed from a hologram is perfectly zero-loss filtered [40]. At the inelastic process, considered as a quantum mechanical measuring process, the previous electron wave function collapses. Nevertheless, after scattering, the inelastic electron is again a wave, newly-born in the scattering event, incoherent with the residual elastic waves. Of course, in the far field of the object, this wave could again contribute to an interference pattern, however, because of broadening of the energy spectrum of the electron ensemble, now temporal coherence is reduced. The basic question remains, which extension the new-born wave has in the scattering volume in the object. For answering this, EFTEM-holography has been conducted in that, in the EFTEM-image, adjacent areas are superimposed by a biprism [41–43]. With increasing shear by increasing the biprism voltage one finds a decay of interference contrast, i.e. one measures the distribution of coherence around the scattering event. The reduction of coherence with respect to the initial “ensemble coherence” of illumination depends on the excited state, and therefore we name it “state coherence”. Our investigations reveal that the waves new-born in the scattering event have a mean extension dcoh in the object depending on the energy transfer ΔE suggesting a relation dcoh  1=ΔE consistent with theory [44]. For example, dcoh ≈30 nm was found at excitation of bulk plasmons in Si with ΔE≈16 eV; for surface plasmons with ΔE≈7 eV, dcoh is significantly larger [45]. This holds also for electrons passing an object at some distance, exciting “aloofly” an inelastic event. This means that the coherence area reaches from the object into vacuum, possibly covering also the reference wave. In this case, inelastic interference would contribute to a hologram and hence the reconstructed wave would carry also inelastic information; this is discussed in detail in [46].

3.6. Role of accelerating voltage Early electron microscopy and hence also experiments in interferometry and holography have mostly been performed at low acceleration voltages of only several 10 kV, because mastering of higher voltages was difficult. However, for achieving atomic resolution with an aberrated TEM, acceleration voltages of several 100 kV turned out indispensable hence were made available by TEM-manufacturers. Alas, this improvement of resolution had to be paid with knock-on radiation damage in particular of light elements (see e.g. [47–50]). With availability of aberration correctors, atomic resolution became possible also at acceleration voltages as low as 40 kV. In face of the growing interest in light element materials such as carbon, low accelerating voltages are increasingly interesting for TEM-methods [51–54] in order to reduce knock-on beam damage.

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Also for holography, acceleration voltage may be crucial. For example, without hardware correctors but with a-posteriori aberration correction, atomic resolution can only be reached at U acc ≥300 kV because of the information limits set up by incoherent aberrations [55,56]. With accelerating voltage also the extinction thickness increases and hence objects behave kinematically rather than dynamically at larger thicknesses hence allow a better averaging of, say, the mean inner potential. Furthermore, high voltage beams are less sensitive against stray fields and produce less charging effects because of reduced secondary electrons yield. Knock-on damage is enhanced, but ionization damage (radiolysis) is reduced. On the other side, there are several advantages of low-voltage electron holography. Reduction of knock-on beam damage is very important because it ultimately limits the dose accumulated in a hologram hence NoiseFigure; for dose-rate dependent damage processes such as radiolysis, at low current densities exposure time τ can be extended in the limits discussed above. However, radiolysis increases with decreasing voltage. Another major benefit is the gain of phase shift from electric potentials evident from s ¼ e=ðℏvÞ in Eq. 7, as experimentally exploited by [57]; this has been used in early 8 keV-electron holography to measure the phase shift from weak objects such as unstained bacteriophages [58]. However, dynamical diffraction and back-scattering effects get stronger and the phase object approximation may not hold [59]. Interestingly, magnetic phase shift is independent from accelerating voltage at all. In any case, low voltage also enhances the inelastic interaction and hence noise in the reconstructed wave; if this effect dominates Inf oCont, signal/noise ratio is hardly improved by low voltage. The lower biprism voltage improves the stability of biprism action. Additional important benefit stems from significantly improved camera characteristics [60]. For instance, with a state-of-the-art CCD camera the MTF at the normal side-band position is about two times better for 80 kV than for 300 kV. The improvement of the DQE is even more than a factor of three [61]. Further effects of a lower acceleration voltage are a larger end-magnification that improves the sampling—especially important for high-resolution holography—but may also introduce image distortions. At last, when using lower electron energies one has to be more careful in the interpretation of the recorded potentials because the sample may be charged up more easily and thus introduce additional potentials not contained in the original sample structure.

3.7. Bright field–dark field 3.7.1. Bright field holography The standard method is bright field holography. At medium resolution o qmin ; i.e. below the Scherzer band, the reconstructed wave agrees with the object wave hence can be evaluated directly. At higher resolution 4 qmin ; after correction of coherent aberrations, be it by a hardware corrector or by a-posteriori holographic correction, the reconstructed wave comprehensively represents the object exit wave within the incoherent limits of transfer theory of the TEM; also with a hardware corrector, a-posteriori correction is extremely helpful for fine-tuning the residual aberrations [62]. From the resulting atomically resolved wave, all respective properties of the object can be evaluated. All methods of conventional imaging can be realized by image processing from the reconstructed wave both in real space and in Fourier space, such as dark field imaging with amplitude and phase in the light of arbitrary reflections, or holographic nanodiffraction [63]. The magnetic contributions are separated from the electric ones by recording two holograms with reversed objects; the sum of phases gives the electric part, the difference the magnetic part [64].

3.7.2. Dark field holography Hanszen and coworkers investigated the appearance of the waves diffracted at crystals [65]. They found that determining the phase distribution in the dark field wave in real space allows direct access to object properties such as defects and effects of dynamic interaction. For this, atomic resolution is not needed in real space, because the method evaluates the amplitude and phase distribution coined on a diffracted wave arising from atomic details, for example at defects. With coarse hologram fringes the field of view can be made comparably large. The experimental procedure is that one selects the object wave in the distorted crystal area, whereas the reference wave is positioned in a perfectly grown crystal area. Superposition of the waves forms an interference pattern, which Hanszen called a dark-field hologram. Subsequently, from the sideband of this hologram, amplitude and phase of the diffracted wave can be reconstructed in real space. But as promising Hanszen's approach was, dark-field holography almost was buried in oblivion, until Hÿtch rediscovered the technique for strain measurements in electronic devices. Since the phase of the diffracted wave is proportional to the displacement in a strained lattice, it provides direct access to local changes in the lattice parameter, leading to a two-dimensional strain map [66]. Since exact knowledge about strain fields is crucial in semiconductor device metrology, much research is carried out on checking and improving the method´s reliability [67].

3.8. Variation of holographic recording 3.8.1. Paths of rays: variation of lens optics Interestingly, the relation Inf oCont ¼ nrec nφ holds true independently from the actual size of field of view, be it 1 cm or 10 nm, or the respective fringe spacing, established with appropriate electron optics. It is more a problem of instrumentation rather than physics that fine hologram fringes are much more sensitive against disturbances. Of course, to achieve a special field of view, one needs specific paths of rays combining the lenses for imaging and the biprism for aperture the waves. Usually, the biprism position is the position of the SA holder because of ease of access. For medium resolution, the Lorentz lens used as objective lens allows a field of view of a few mm; this is very useful for the investigation of large areas in semiconductors and magnetic specimen. For a smaller field of view of about 10 nm hence a better lateral resolution reaching to about 0.1 nm, the objective lens is much superior, however, with the object embedded in the strong lens field of about 2 T. By specific excitations of objective lens or Lorentz lens and of the subsequent intermediate lens one can cover a range of field of view reaching from some mm down to several nm at corresponding resolutions given by nrec for nearly every purpose [68]. For atomic resolution, the biprism is best inserted close to an intermediate image plane at about 550 times magnification [69]. This special arrangement cannot be realized with a standard microscope; however, it is realized in a microscope specially designed for the purpose [70]. In general, a microscope equipped with an aberration corrector offers additional degrees of freedom. In [71], Snoeck et al. report about the use of the first transfer lens of the aberration corrector as a pseudo-Lorentz lens for investigating magnetic materials. Whereas, at first, the hexapoles have been switched off during Lorentz mode operation, they finally allow correcting the giant spherical aberration of the conventional Lorentz lens, and hence in particular promise to increase spatial resolution in Lorentz mode [72]. In addition, combining the aberration corrector in high resolution or Lorentz mode operation with specific excitations of the subsequent intermediate lens enables a comprehensive holographic analysis at nearly all magnifications within one TEM column [73].

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3.8.2. Paths of rays: using several biprisms Using only one biprism, there is, besides lens parameters, only one degree of freedom, i.e. the biprism voltage, to control both the width and the fringe spacing of the hologram; even worse, the reference wave has to be selected from an area closely adjacent to the object area of interest, because both must lie well inside the coherently illuminated patch. This restricts the accessible areas of the object to the edge or close to a hole; there, however, the object may have properties different from the bulk. Furthermore, it is sometimes very difficult to exclude the effects of stray fields from the object laterally leaking out into the reference wave. For more flexibility it would be ideal, to split the reference wave and the object illuminating wave before the object like in a light-optical Mach–Zehnder interferometer, and to navigate them independently through the microscope column. This can be achieved by the use of several biprisms arranged in the optical path as introduced by Möllenstedt and Bayh in their famous 3-biprism interferometer [74]: the first biprism before the object splits the illumination into a reference wave and an object-illuminating wave; by the negative filament voltage, these two waves diverge from each other and pass the object plane at a freely selectable mutual distance. The coherence, carried along with the waves, is in principle conserved; however, one needs special care, because at a large mutual distance the two widely separated waves become increasingly sensitive against vibrations introducing statistical phase differences, and, because of the ESAB-effect, the larger area between them makes them more sensitive against AC-stray fields; at the end, these effects destroy coherence between the waves. The subsequent biprisms behind the object, together with the lenses, superimpose the two waves in a hologram of selectable width and fringe spacing in the detector plane. The same principle was used in the electron mirror interference microscope [75] and in the special interferometer reaching a separation of 300 mm between the coherent waves [76], built e.g. for accommodating an encapsulated coil between the waves for investigating the ESABeffect in more details. Meanwhile, also TEMs dedicated for holography are increasingly conceived for accommodating multiple biprisms. These offer the advantage of freedom from Fresnel modulation in the hologram, and of a higher degree of freedom for width of hologram, fringe spacing and fringe orientation [77–79], and a large deflection angle at small biprism voltages, in particular at high acceleration voltages of 1 MV [56]. In [80], a biprism is incorporated in the condenser optics to split the illumination, and three others follow to overlap the split waves in the image plane. A similar scheme is reported in [81].

3.8.3. Other arrangements In the previous sections we discussed the creation of holograms in the image plane if the specimen is illuminated with plane illumination. This is but one particular optical setup for performing electron holography. A large diversity of other setups differing in both illumination and interference plane have been proposed and experimentally realized [23]. For example CBED holographic experiments have been reported by Pozzi at al. [82,83], Möllenstedt et al. [84], and most recently by Houdellier and coworkers [85]. In this setup the specimen is illuminated with convergent illumination and the hologram is formed in the Fourier plane of the specimen. As a consequence, this setup facilitates the reconstruction of the complex aberration function as well as complex structure factors. Another example is STEM holography: by means of a biprism in the condenser optics, two coherent STEM-probes are generated. These two probes are scanned synchronously, one across the object and the other one across a reference area. Behind the object plane,

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the two waves are de-scanned. Therefore, in the detector at far field, the two corresponding waves overlap in a stationary interference pattern shifted aside only if a phase shift occurs from the object. The detector compares the position of the fringes with a fixed grating of same period and creates a phase and an amplitude image on the STEM-monitor [86]. 3.9. Holographic tomography A phase image contains only the projected 2D information about the specimen, whereas the local distribution along the electron beam, i.e. along the direction of projection, is lost. Thus, the quantitative mapping of electric or magnetic potentials is only possible, if additional information (or assumptions) about object thickness and homogeneity can be provided. In any case, only the potential averaged along the electron beam can be extracted. This problem can be solved by combining electron holography (EH) with electron tomography (ET) into electron holographic tomography (EHT): to this end, a tilt series of electron holograms must be acquired in the TEM with a tilt of the specimen under the electron beam at commonly 7701 and increments of 1−31. From the respective electron holograms, the single 2D-electron waves are reconstructed yielding both an amplitude and phase tilt series. Finally, the tilt series of the 2D-phase images, i.e. the projected potentials, is used to obtain the 3D-potential by tomographic reconstruction techniques. A tilt series of phase images is a very suitable dataset for tomography because a phase image can be considered in Phase Grating Approximation as projected potential. In 1994, Lai and co-workers published two articles describing the proof-of-concept in which they showed that EHT enables the reconstruction of the electrostatic potential [87] as well as the magnetic field [88]: they carried out 3D reconstructions of the electrostatic potential in three adjacent latex spheres (ca. 120 nm diameter) as well as one component of the magnetic induction outside a barium ferrite particle (ca. 1 μm in size). However, this work was not yet quantitative, i.e. they did not determine values for the reconstructed electric potential and magnetic field. 10 years later, Friedrich et al. illustrated that the 3D phase map of magnetotactic bacteria can be quantified in terms of the mean inner potential distribution within the sample [89]. More recent studies of Twitchett–Harrison et al. extended this approach to the analysis of the 3D electrostatic potential distribution across p–n junctions in silicon [90,91]. The thereby achieved resolution was about 20 nm, laterally in x and y. Very recently, Tanigaki et al. showed that they could resolve features with a size of about 2 nm within reconstructed 3D potentials of platinum nanoparticles on an amorphous silicon pillar [92]. In the Triebenberg Lab., we have developed the demanding and time-consuming EHT workflow to a fully quantitative and widely automated method. The automation involves a dedicated software package for acquisition (THOMAS— tomographic and holographic microscope acquisition software) [93,94] and reconstruction of an entire holographic tilt series. This represents an efficient tool to reconstruct in 3D the mean inner potential (MIP) distribution yielding the inner and outer structure of nanostructures, e.g. of GaAs/AlGaAs core–shell nanowires [95], or the distribution of functional potentials, e.g. built-in potentials across p–n junctions in Ge [96], with high accuracy and within a reasonably small amount of time (1–2 days). The ability of EHT to provide quantitative insight into the electrostatic 3D potential structure is unique and beyond the capabilities of conventional ET in bright field (BFTEM) or scanning (STEM) imaging mode. 3.10. Reconstruction of the object wave In the hologram, the wave Aðx; yÞexpiϕðx; yÞ is encoded in the interference pattern of cosinoidal fringes modulated in contrast by

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the amplitude Aðx; yÞ and in position by the phase ϕðx; yÞ. The fringes are the carrier of information. Reconstructing the wave means lifting it off the carrier as accurately as possible. After all, the initial procedure by Wahl still seems to be the best way. It is described in detail, for example, in [24]. The Fourier transform consists of a centerband and two conjugate complex sidebands. The +1-sideband is the Fourier Transform of the desired wave. Since it is considered band-limited to a maximum spatial frequency qresmax by a mask in the back focal plane of the TEM or by the incoherent envelope function, the center band has an extension of 2qresmax . Consequently, to avoid overlap, a carrier frequency qc ≥3qresmax separating sideband from centerband is needed; for pure phase objects qc ≥qresmax may be sufficient. Then the sideband can completely be isolated by means of a mask, centered and inversely Fourier transformed to real space. Size and shape of the mask determines both lateral and signal resolution, corresponding to the above discussed Inf oCont ¼ nrec nφ . Therefore, the mask must be optimized for the specific question one wants to answer. For example, stray fields, given by the Laplace equation, are slowly varying and mostly weak, and hence a small mask reducing noise at the cost of lateral resolution could be favorable. To avoid in real space a point spread function with far-reaching side-lobes, mask with soft edges, for example with a Gaussian profile or a Hanningwindow, can be helpful. In any case, the reconstruction method in Fourier space is a global method in that all pixels in real space are treated in the same way. This may be disadvantageous, for example if there are strong amplitude modulations affecting the noise locally. Therefore, schemes for local reconstruction have been conceived: a small aperture is slid across the hologram and in every position fringe contrast and fringe position are evaluated [97,98]; this is equivalent to using a sinc-mask in Fourier space for cutting out the sideband [96]. Local reconstruction would allow applying a local weighting function to cope with noise variations in the hologram approaching a Maximum-Likelihood method. This will be most interesting if the hologram fringe contrast and the camera properties are significantly better than today. 3.11. Evaluation in terms of the object Evaluating wave functions reconstructed by means of off-axis holography in terms of physical quantities of the specimen is generally simplified by the always linear transfer of waves because of the wave equation. This is also valid for holography in the TEM. It means that (in-)coherent aberrations act as a simple convolution on the object exit wave which can either be inverted or taken into account within a certain scattering model without leading to ambiguities such as nonlinearities. In addition, amplitude and phase often allow a straightforward interpretation in terms of meaningful physical quantities. This is true as long the phase grating approximation ϕðx; yÞ ¼

e e V ðx; yÞ− Φðx; yÞ ℏv proj ℏ

ð15Þ

holds for the range of interaction conditions, where the specimen is illuminated under directions well away from low-index zone axes. Similarly, Z lnðAðx; yÞÞ ¼ − ρðx; y; zÞ sinel ðx; y; zÞ dz ð16Þ holds for a wide range of imaging conditions, with ρ the particle density and sinel the total inelastic scattering cross section for one particle. Usually, the integral is used in the abbreviated form t=λinel with thickness t and mean free path for inelastic interaction λinel [99]. These expressions even remain valid under slight dynamic conditions by incorporating a specific correction factor [100]. Likewise,

the comparably weak functional potentials in a solid, e.g. in p-n junctions, may be regarded as a weak addition to the total potential; for these functional potentials the above relations may hold even under dynamical conditions for the crystal potential [101]. However, for imaging conditions with large influence of dynamic scattering such as low-index zone axes used for high resolution holography, the object exit wave does not allow a simple connection to the scattering potential, and the interpretation of reconstructed waves is rather involved. In this case, a fully dynamical calculation has to be compared with the experimental findings. Furthermore, the simple line integral structure of the phase shift facilitates a straight forward calculation of charges and magnetic moments as source terms for electric and magnetic fields. This follows directly by inserting static Maxwell's equations into the projection integrals. For example in case of the electrostatic potential, integrating the 2D Laplacian Δ⊥ of the phase over a certain region, directly yields the enclosed charge Q according to 2

3

∬ Δ⊥ φðx; yÞd R ¼ C E ∭ Δ⊥ Vðx; y; zÞd r lim Ez ðx;y;zÞ ¼ 0

3

C E ∭ ΔVðx; y; zÞd r CE Gauss0 s law C E 3 − ∭ ρðx; y; zÞd r ¼ − Q : ¼ ε0 ε0 z-∞

¼

ð17Þ

This can be used to measure charges on nanoparticles [102] and nanowires [103] with a precision in the order of a single elementary charge. Similar considerations also hold for the magnetic moment. The projected magnetic field is obtained from ! ℏ! B p ðx; yÞ ¼ z u  ∇ϕðx; yÞ ð18Þ e ! with z u being the unit vector-in beam-direction, and it can be shown that the integration of B p over a disc gives the enclosed magnetic moment according to [104] 2 ! ! ∬ B p ðx; yÞ dx dy: m¼ μ0

ð19Þ

Due to the much smaller magnetic phase shift, the currently achieved precision is in the order of 104 Bohr magnetons. Please note that this can only be achieved by integrating over large areas hence at cost of lateral resolution. This again emphasizes the concatenation of lateral and signal resolution in the presence of noise.

4. Conclusion After more than 40 years of development, electron holography has reached the performance, i.e. lateral resolution and signal resolution, which allows detailed and quantitative investigation of electric and magnetic nanofields in solids down to an atomic scale. By carefully optimizing the recording, noise is suppressed and hence information content InfoCont is further improved. Novel instrumentations for recording holograms of arbitrary object areas with a really undisturbed reference wave are surmounting the experimental restrictions. Even if we had solved all present shortcomings of the hitherto available holographic methods and succeed in reconstructing a perfect wave, the most severe point will be the unique interpretation of the findings in term of the object. The reason is that very many different properties of an object are mixed together in one quantity, in particular in the phase, and it will be a huge challenge to identify and evaluate each of them separately. As shown above, the magnetic phase shift can be distinguished hence be separated from the electric ones. But there is a whole bunch of object characteristics, which generate electric phase shifts:

 Mean Inner Potentials (MIP)  local variations of MIP at defects, boundaries, e.g. due to strain

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surface states trapping charges surface reconstruction surface adsorption doping, intended or unintended (impurities) electric polarization, e.g. in ferroelectrics or in polarized nanostructures such as LaAlO3 interface segregation, outdiffusion interdiffusion in heterogeneous stacking layers mobile charges compensating inner fields thickness variations, e.g. by preferential etching contact potentials, Schottky effect Charging under the beam Electron beam induced currents and voltage drops

Besides, the more accurate the data are, the more one also has to understand the contributions from residual phase shifting effects stemming from dynamic interaction. Therefore, we have to strengthen in-situ experiments where only the parameter of interest is varied, and a better insight in solid state science has to be gained and used for a more detailed and accurate modeling of the substance under investigation.

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