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Journal of Electron Spectroscopy and Related Phenomena 178–179 (2010) 195–220

Contents lists available at ScienceDirect

Journal of Electron Spectroscopy andRelated Phenomena

journa l homepage: www.e lsev ier .com/ locate /e lspec

hotoelectron holography with improved image reconstruction

omohiro Matsushita a,∗, Fumihiko Matsui b, Hiroshi Daimon b, Kouichi Hayashi c

Japan Synchrotron Radiation Research Institute (JASRI), SPring-8, 1-1-1 Kouto, Sayo-cho, Sayo-gun Hyogo 679-5198, JapanNara Institute of Science and Technology (NAIST), 8916-5 Takayama, Ikoma, Nara 630-0192, JapanInstitute for Materials Research, Tohoku University, Sendai 980-8577, Japan

r t i c l e i n f o

rticle history:

a b s t r a c t

vailable online 18 June 2009

eywords:hotoelectron holographyuger electron holography

Electron holography is a type of atomic structural analysis, and it has unique features such as elementselectivity and the ability to analyze the structure around an impurity in a crystal. In this paper, we intro-duce the measurement system, electron holograms, a theory for the recording process of an electronhologram, and a theory for the reconstruction algorithm. We describe photoelectron holograms, Augerelectron holograms, and the inverse mode of an electron hologram. The reconstruction algorithm, scat-tering pattern extraction algorithm (SPEA), the SPEA with maximum entropy method (SPEA-MEM), and

onal
orward focusing peak SPEA-MEM with translati
. Introduction

Since the beginning of the 21st century, nanoscale semicon-uctors, superconductors, superfine particles, magnetic molecules,iomolecules, functionalized molecules, etc. have been researchedxtensively. These developments have been supported by nanoscaleeasurement technology such as mass analysis, scanning tun-

eling microscope, and synchrotron radiation based technology.esearches that require information of an atomic level are growingow.

Electron holography is a method used for atomic structural anal-sis. Many researchers have studied electron holography because ofts unique features. (1) It has good surface sensitivity. (2) The localtomic structure around a target atomic site is observable. (3) Theocal atomic structures around an impurity in a crystal, or adsorbaten a crystal, which have no long-range order, can be observed. (4)

nitial information about the atomic arrangement (model or phase)s not required.

The principles of electron holography were proposed by Gabor1] in order to improve the electron microscope. He pointed out thathe interference between an object wave and an reference wavean be produced by using a focused electron beam that acts as aoint source. The size of the point source affects the spatial reso-

ution; the spatial resolution improves with a reduction in the size
f the point source. Szöke pointed out that photoelectrons, Augerlectrons, and X-ray fluorescence emitted from an atom can acts atomic-size point sources. Subsequently, photoelectron hologra-hy (PEH) (based on the use of photoelectrons), defuse low-energy
∗ Corresponding author.E-mail address: [email protected] (T. Matsushita).

368-2048/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.elspec.2009.06.002

operation are also described.© 2009 Elsevier B.V. All rights reserved.

diffraction (DLEED) (based on low-energy diffraction (LEED)) [2],Kikuchi electron holography (KEH) (which utilizes the Kikuchi elec-trons) [3,4], etc. have been proposed. Recently, the inverse modeof photoelectron holography [5] that utilizes the time reversal ofnormal holography has been proposed. In this paper, these atomicresolution electron holography techniques are hereafter referredto electron holography. X-ray fluorescence holography (XFH) andits inverse mode (IXFH) have also been developed, recently; thesetechniques can achieve atomic resolutions [6–14]. Atomic resolu-tion holography techniques that are being developed currently useeither electrons or photons. In this paper, we mainly describe elec-tron holography.

An electron hologram is basically measured as follows. A sam-ple is irradiated using a soft X-ray. An electron is then excitedfrom the atom. A part of the excited electron wave is scatteredby the surrounding atoms. Unscattered electron waves and thescattered waves are interfered. The unscattered waves and thescattered waves act as reference waves and object waves of holog-raphy, respectively. The angular distribution of the electron formsthe hologram. Electron holography is surface sensitive because thelength of the mean free path of the electron in the solid is of theorder of a few nanometers. A target atomic site can be selectedby using the core binding energy since this energy is different foreach element, and it depends on the electronic environment in asolid (chemical shift). It is possible to measure an electron hologramaround a target atomic site. In addition, an impurity in a crystal oran adsorbate on a crystal surface, which have a no long-range order,
can be selected as the target atomic site.
A three-dimensional atomic image cannot be directly observedfrom the electron hologram by using a method similar to thatused for an optical hologram because the conditions of the scat-tering process and interference are different. The atomic image is

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1 opy and Related Phenomena 178–179 (2010) 195–220

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ltmsm[sA

2

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Fig. 1. Principle of electron holography. (1) Excitation light is irradiated on a crystal.

96 T. Matsushita et al. / Journal of Electron Spectrosc

econstructed by using a reconstruction calculation. The recon-truction algorithm was proposed by Barton in 1988 [15]. Thislgorithm is based on the Fourier transform; however, a cleartomic image could not be obtained by using this algorithm becausehe effect of the phase shift of the scattering process is neglected. Inddition, a conjugate image, which is a virtual image located on theoint symmetric position, appears. In order to solve the problem ofhe conjugate image, an electron hologram having a multi-energyormat was proposed [16]. Although this enabled the conjugatemage problem to be solved, the effect of the phase shift could note solved. The typical phase shift problem is the forward focus-

ng effect, a phenomenon in which electrons are focused towardhe direction of the scatterer atom. The forward focusing peakauses image aberrations in the reconstructed atomic image. Sub-equently, improved algorithms based on the multi-energy formatere studied [17–23]. For example, reconstruction algorithms such

s scattered-wave-included Fourier transform (SWIFT) [17,18] andmall-window energy-extension process (SWEEP) [19–21] wereroposed. However, these algorithms did not always provide aood atomic image [24]. Differential holography, which utilizes thenergy differential of the photoelectron hologram, was also pro-osed [25]. This method can remove the forward focusing effecty using the energy differential of the photoelectron hologram,ecause the energy dependence of forward scattering is weakerhan that of back scattering. The effectiveness of this method wasnvestigated. It was found that the measurement of a highly preciseifferential photoelectron hologram may require a long time. Nearode holography was proposed as well; this experimental methodan remove the forward focusing effect by measuring photoelec-rons near the node of the emission angle of the photoelectrons [26].lthough this method was found to be effective, one must measureeak emissions of the near node of the s initial state. The initial

tates of p, d, or f cannot be utilized. The cross section of s initialtate is smaller than that of p, d, or f initial states and the emissionear the node is very weak. Therefore, this method may require

ong measurement time. In addition, these measurement methodsre based on the multi-energy format. Since the multi-energy for-at requires an energy-tunable light source (synchrotron radiation,

tc.), it is not facile. In addition, single-energy holograms such asuger electron holograms are inapplicable.

We developed a reconstruction algorithm that does not uti-ize the Fourier transform. This algorithm solves the problem ofhe phase shift and the conjugate image, and the atomic arrange-

ent can be obtained without using the multi-energy format. Weucceeded in reconstructing a three-dimensional atomic arrange-ent from a single-energy hologram with a resolution of 0.02 nm

27–30]. This algorithm enables the atomic arrangement to be mea-ured from a photoelectron hologram using an X-ray tube or anuger electron hologram.

. Recording process of electron holography

Electron holography consists of the recording and reconstruc-ion processes. A schematic view of the recording process of thelectron holography is shown in Fig. 1. The recording process ofhe electron hologram is described as follows. (1) A photoelectronr Auger electron is excited by using monochromatic light. (2) Thexcited electron wave propagates as a spherical wave. (3) A partf the spherical wave is scattered by surrounding atoms formingscattered electron. The part of the wave that remains unscat-

ered becomes the reference wave, and the scattered and referenceaves interfere with each other. (4) The two-dimensional angularistribution of the intensity of electrons having a kinetic energy EkI(Ek, �,�)) can be regarded as a single-energy hologram. When theinetic energy dependence is measured, a three-dimensional inten-

(2) A photoelectron or Auger electron is excited. (3) A part of the electron waveis scattered by the surrounding atoms forming a scattered wave. (4) The scatteredand direct waves interfere, and the interference pattern is observed as the angulardistribution of the electron. This pattern can be treated as the hologram.

sity distribution I(Ek, �,�) can be obtained. This is a multi-energyformat of the hologram.

2.1. Measurement of the electron holography

Electron holograms can be measured easily because the ampli-tude of the electron hologram is over 10% of that of the referencewave. It is recommended that electron holograms are measuredusing kinetic energies of over 400 eV. There are three reasonsfor this. The spatial resolution improves with increasing kineticenergy. The electron path is refracted at the surface of the solidbecause of the inner potential. When the kinetic energy increases,the refraction effect become negligible. The scattering pattern func-tion, which is discussed below, has a simple shape when the kineticenergy is over 400 eV. However, the surface sensitivity decreaseswith increasing kinetic energy. Therefore, it is recommended thatthe electron holograms are measured under the highest electronintensity from the target element when using a kinetic energy ofover 400 eV. In order to utilize the above-mentioned reconstructionalgorithm, which is described below, the two-dimensional angulardistribution of the electron on the 2� steradian is required. Therequired accuracy of the angle is approximately to 1◦.

Various measurement methods have been proposed [31–35].The most popular method is the combination of an angle-resolvedelectron energy analyzer and a sample manipulator that can sweepangles � and� of the sample. Another method is a two-dimensionaldisplay-type analyzer [35]. This apparatus can project an angulardistribution of ±60◦ on the screen.

2.2. Energy dispersive electron energy analyzer

A relatively easy way to measure the electron hologram is touse the commercially available electron analyzer and a manipulatorthat can sweep angles � and � of a sample. A schematic view of themeasurement apparatus is shown in Fig. 2. We also constructed thistype of apparatus using SPECS Phoibos 150 and an X-ray tube [27].

We measured the photoelectron holograms of Si(0 0 1) and
Ge(0 0 1) by using this apparatus in order to examine the above-mentioned reconstruction algorithms. The photoelectron hologramof Si(0 0 1) 2s having a kinetic energy of 1335.5 eV was measuredby using Al-K� light. The measured hologram is shown in Fig. 3.A clear interference pattern was measured. It is possible to mea-

T. Matsushita et al. / Journal of Electron Spectroscopy and Related Phenomena 178–179 (2010) 195–220 197

Fe

smehlTmbamsrti

2

msvtpA

Fro

Fig. 4. A photoelectron hologram of Ge(0 0 1). Ge 3s initial state was excited bycircularly polarized synchrotron radiation. The kinetic energy is 620.5 eV. A fourfoldsymmetry operation was applied.

ig. 2. Experimental setup of the electron hologram using a well-known electronnergy analyzer.

ure the photoelectron hologram by using an X-ray tube. We alsoeasured a Ge(0 0 1) 3s photoelectron hologram having a kinetic

nergy of 620.5 eV by using synchrotron radiation. The measuredologram is shown in Fig. 4. In these experiments, the energy reso-

ution and angular resolution were set to 0.9 eV and 1◦, respectively.he azimuthal angle�was scanned in the range of 0–180◦. After theeasurement, the obvious noise was removed by using a software-

ased frequency filter, and the rotational symmetry operation waspplied considering the symmetry of the crystal. In this case, theeasurement time required was approximately 4 h. A long mea-

urement time is required because the manipulation of the sampleequires a long time. This is the disadvantage of this type of appara-us. However, this method can achieve high energy resolution, andt enables atomic site separation by using the chemical shift.

.3. Two-dimensional display-type spherical mirror analyzer

A two-dimensional display-type spherical mirror analyzer caneasure an electron hologram very rapidly. We developed and con-

tructed such an analyzer at BL25SU in SPring-8 [36]. A schematic
iew of the apparatus is shown in Fig. 5. It is possible to measurehe electron hologram with a cone angle of ±60◦ at once. An exam-le of a measured image is shown in Fig. 6. This image shows anuger electron hologram of Cu(0 0 1) L3VV [29]. This clear image
ig. 3. A photoelectron hologram of Si(0 0 1). Si 2s initial state was excited by Al-K�adiation. The kinetic energy of the photoelectron is 1335.5 eV. A fourfold symmetryperation is applied.

Fig. 5. Two-dimensional display-type analyzer developed at BL25SU in SPring-8.

was obtained by normalizing the measured raw image by the trans-
mission function of the analyzer. In the case of the Auger electron ofCu(0 0 1), it is possible to perform the measurement within an expo-sure time of 0.1 s. Therefore, real-time measurement is possible [37].When the incident angle of light is set to be normal and the sam-
Fig. 6. A measured image of the L3VV Auger electron hologram of Cu(0 0 1) by usingthe two-dimensional display-type analyzer. The kinetic energy is 914 eV. The imagewas normalized by the transparent function of the analyzer.

198 T. Matsushita et al. / Journal of Electron Spectroscopy and Related Phenomena 178–179 (2010) 195–220

FT

pfbptm

2

aopTeritttiFi

Fce

scattering pattern function.When the excitation light is irradiated on the sample, a photo-

electron or an Auger electron is excited as shown in Fig. 1. An atomthat emits an electron called as emitter. Here, the muffin-tin poten-tial radius of the emitter is defined as b. Since the region where r > b

ig. 7. An L3VV Auger electron hologram of Cu(0 0 1). The kinetic energy is 914 eV.his image was developed from the 36 measured images.

le is rotated in a plane, an electron hologram of the hemisphereace is obtained. An Auger electron hologram of Cu(0 0 1) measuredy using this apparatus is shown in Fig. 7[29]. Figs. 8 and 9 showhotoelectron holograms of Si(0 0 1) [38] and Cu(0 0 1) [30], respec-ively. Clear Auger electron and photoelectron holograms could be

easured.

.4. Inverse mode electron holography

The normal electron scattering process has been describedbove. However, it should be noted that the hologram can also bebtained by using the time-reversal mode of the normal scatteringrocess [5]. A schematic view of this situation is shown in Fig. 10.he apparatus used for this hologram measurement consists of anlectron gun and an X-ray detector for the fluorescence. The fluo-escence X-ray intensity I depends on the electron energy Ek and itsncident angles � and � because the electron probability density onhe internal detector atom is formed by the interference between
he direct wave from the electron gun and the wave scattered byhe surrounding atoms. The intensity distribution I(Ek, �,�) of thencident angles and the kinetic energy forms an electron hologram.or example, the experimental result for a SrTiO3 crystal is shownn Fig. 11. An interference pattern was successfully observed. The
ig. 8. A photoelectron hologram of Si(0 0 1). The Si 2p initial state was excited byircularly polarized light. The incident direction of the light is (0 0 1). The kineticnergy of the photoelectron is 800 eV.

Fig. 9. Measured Cu(0 0 1) PEH. The kinetic energy of the photoelectron is 818 eV,and the initial state is the Cu 3p state. The lines in the figure indicate the Kikuchiband.

inverse mode hologram of an electron hologram is equivalent to anormal hologram obtained by using the electron wave function ofthe s wave.

2.5. Theory of electron holography

The theory of the recording process of electron holography isimportant for the theory of the reconstruction process of the atomicarrangement. Here, we describe the recording process of electronholography. Then, the scattering pattern function is derived. Theabove-mentioned atomic image reconstruction algorithm uses the

Fig. 10. Internal detector electron holography. The incident electron beam reachingthe detector atom either directly or after scattering acts as a holographic reference orobject wave, respectively. The intensity of the characteristic X-rays from the internaldetector atom depends on the total field. Monitoring the characteristic X-rays in theincident electron beam directions maps out the hologram.

T. Matsushita et al. / Journal of Electron Spectroscopy a

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cts

ϕ

k

ioY�v

ϕ

iw

ϕ

Twf

ϕ

hif

ϕ

ϕtd

r

h� is the photon energy. Ei and Ef are the energies of the initialand final states, respectively. The kinetic energy of the final state Efis an important parameter in photoelectron holography since it isrelated to the wavenumber of the electron. When the photon energychanges, the kinetic energy of the final state changes because of Eq.

ig. 11. Hologram of SrTiO3using Ti K � line obtained by internal detector. The holo-ram around the (0 0 1) direction was not able to measure because of the structuref the apparatus used at that time.

an be regarded as the free space, the excited electron wave func-ion can be described by the spherical wave functions in the freepace.

L(k, r, �,�) = k∑lm

ALlmil+1h(1)

l(kr)Ylm(�,�). (1)

is wavenumber given by k =√

2mEk/h. ALlm is a coefficient, and L

s an index for the multiple excited states.h(1)l

(kr) is Hankel functionf the first kind, and it describes the traveling wave from the emitter.lm(�,�) is a spherical harmonic. Here, it is noted that angles � andof the polar coordinate can be regarded as the angle of the motion

ector.

L(k, r, �,�) = ϕL(k, r) = ϕL(k, r). (2)

For example, when the emitted wave function is s wave, thendex set to be L = s, and the coefficient set to be As00 = 1. This

ave function is given as

s(k, r, �,�) = 1√4�

eikr

r. (3)

Then this wave is scattered by the surrounding atoms (scatterer).he wave function of the scattered wave is defined as L(k, r,a),here a is the position vector of the scatterer. The scattered wave

unction is given as

L(k, r) +∑h

L(k, r,ah). (4)

is an index of the scatterer atom. The square of this wave functions observed at infinite distance. The wave functions are defined asollows:

L(k) ≡ limr→∞

r e−ikrϕL(k, r), (5)

L(k,a) ≡ limr→∞

r e−ikr L(k, r,a). (6)

L(K) and L(k,a) are corresponding to the reference wave and
he object wave, respectively. Hankel function at large distance isescribed as
lim→∞h(1)l

(x)∼ (−i)l+1 eix

x. (7)

nd Related Phenomena 178–179 (2010) 195–220 199

The reference wave at large distance is given as

ϕL(k) =∑lm

ALlmYlm(�,�). (8)

The observed intensity is given as

I(k) =∑L

∣∣∣∣∣ϕL(k) +∑h

L(k,ah)

∣∣∣∣∣2

. (9)

This equation describes the interference between the referencewave and the object wave. The hologram function �(k) is definedby removing the reference wave intensity I0(k) as

�(k) = I(k) − I0(k). (10)

The reference wave intensity is described by the sum of themultiple excited states as

I0(k) =∑L

|ϕL(k)|2. (11)

Therefore,

�(k) =∑L

[∑h

(2Re[ϕ∗

L(k) L(k,ah)] +∣∣ L(k,ah)

∣∣2)

+∑h /= h′

∗L(k,ah) L(k,ah′ )

⎤⎦ . (12)

This is the basic equation for a hologram. The first term describesthe interference between the reference wave and the scatteredwave, which is specific for a hologram. The second term describesthe amplitude of the scattered wave. The third term is the interfer-ence between the scattered waves caused by different atoms.

2.6. Photoelectron emitter

In photoelectron holography, inner shell photoelectrons areused. With regard to transition process of a photoelectron, theenergy conservation law is important.

h� = Ef − Ei, (13)

Fig. 12. Energy diagram of the photoexcitation process.

2 opy a

(teTEeesa

eettsi[

btire

�

Rebt

wrfRtH

˚

ıbbepc

ϕ

�

�vs


13). Therefore, one can obtain a multi-energy hologram. The rela-ionship between the energy levels is shown in Fig. 12. The referencenergy for the initial state is usually set to be the Fermi energy EF .he binding energy EB is usually defined to be the positive, andi = −EB is also defined. The binding energy EB is specific for anlement, and the energy level shifts depending on the chemicalnvironment of the element (chemical shift). Therefore, an atomicite can be identified from the photon energy and kinetic energy ofn emitted electron.

Here, the measured kinetic energy is denoted by Ek; the refer-nce energy is usually set as the vacuum energy EV . The kineticnergy Ek is given by Ek = h�− EB − �, where � is the work func-ion. Here it should be noted that in the case of electron holography,he interference process occurs in a solid. The kinetic energy in theolid does not originate from the vacuum level. The kinetic energyn solid Ek is given by Ek = Ek + V0,where V0 is the inner potential39]. The inner potential V0 is approximately about 8–20 eV.

The excitation process of photoelectrons was described by Gold-erg et al. in detail [40]. The described equations directly providehe wave function at a large distance from the emitter. However, its difficult to apply the equation to the scattering theory because itequires the wave function at a small distance. Here, the followingquations are described.

The initial state is defined as

L(r) ≡∑lm

nLlmRl(r)Ylm(�,�). (14)

ˆ l(r) and nLlm are the radial function of the initial state and thexpansion coefficient, respectively. L is an index used to distinguishetween the initial states. For example, in the case of p initial states,hree states L = px, py, pz should be considered.

The wave function of a photoelectron is defined as follows. Theave function inside the muffin-tin sphere of the emitter atom< b is influenced by the potential of the emitter atom. There-

ore, the radial function inside the muffin-tin sphere is defined asl(kr). The radial function of the wave function outside the muffin-in sphere, i.e. in the free space, is given by the solution of theelmholtz equation. Therefore, the basis function is described as

lm(k, r) =(Rl(k, r)Ylm(�,�) r < b

eiıl il+1kh(1)l

(kr)Ylm(�,�) r ≥ b

). (15)

ˆl is the phase shift of the excitation process, which is determinedy the smooth connection to the radial function of a Rl(k, r) at theorder of the muffin-tin sphere. The wave function of the emittedlectron is expanded by using this basis function. According to theerturbation theory, the wave function of the emitted electron isalculated as

L(k, r) = −2�i∑lm

˚lm(k, r)〈˚lm(k, r)|� · r|�L(r)〉. (16)

Here, the dipole transition operator is given by

· r = �x xr

+ �y yr

+ �z zr. (17)

x, �y, and �z are the x, y, and z components of the polarizationector, respectively. Writing the position operators in terms of thepherical harmonics,

x√

4�(−Y + Y )

r=

311 1−1√

2, (18)

y

r= i√

4�3

(Y11 + Y1−1√

2

), (19)

nd Related Phenomena 178–179 (2010) 195–220

z

r=√

4�3Y10. (20)

Here, the polarization vector is redefined as

�1 = −�x + i�y√2

, (21)

�0 = �z, (22)

�−1 = �x + i�y√2

. (23)

�±1 describes the component of the circularly polarized light. Thedipole transition operator can be written as

� · r =√

4�3

1∑m′′=−1

�m′′Y1m′′ . (24)

Then, the part of the transition probability is given by

〈˚lm(k, r)|� · r|�L(r)〉

=∑l′m′Rll′′ (k)nLl′m′

1∑m′′=−1

�m′′c1(l,m, l′m′)ı(m−m′ −m′′), (25)

Rll′ (k) =∫ b

0

Rl(k, r)Rl′ (r2)r2 dr. (26)

c1(l′,m′, l,m) is the Gaunt coefficient. Fig. 13 shows the results ofthe angular distribution of photoelectrons calculated using Eq. (11).The intensity of the photoelectrons is enhanced along the polar-ization vector, and it is weak along the orthogonal orientation ofthe polarization vector and the direction of the light. When theinitial state is different, the angular distribution of the photoelec-trons is also different. In the case of circularly polarized light, thephotoelectron intensity is strong for an orthogonal orientation oflight.

In the case of photoelectron holography, the anisotropy of thephotoelectron causes problems in the reconstruction calculation.Therefore, non-polarized light is the most suitable option. Circu-larly polarized light is next best option. In the case of circularlypolarized light, an excited wave becomes a spiral wave, and thespiral wave changes the pattern of holography (Daimon effect) [41].The reconstructed atomic image becomes distorted because of thiseffect. However, this effect is limited.

2.7. Auger electron emitter

When an electron is removed from a core level of an atom, anelectron from a higher energy level may fall into the vacancy, result-ing in a release of energy. This energy is transferred to anotherelectron that is then ejected from the atom. This second ejectedelectron is called an Auger electron. The kinetic energy of the Augerelectron corresponds to the difference between the energy of theinitial electronic transition and the ionization energy of the elec-tron shell from which the Auger electron was ejected. These energylevels depend on the type of atom and the chemical environment.The kinetic energy of an Auger electron can be used to determinethe identity of the emitting atom.

In the case of an Auger electron, the cross section may belarger than that of a photoelectron, and the detection efficiencymay be better. However, the kinetic energy is element specific,
and it is difficult to measure a multi-energy hologram. The Augerelectron is usually emitted isotropically; however, the wave func-tion of the Auger electron is not always an s wave. This dependson the transition process of the Auger transition. For example, ithas been reported that the Auger electron of Cu L3VV is close to


Fig. 13. The angular distribution of the excited photoelectron. The polarization of light is indicated by the arrow E. (a) The case of the s state excited by linearly polarized light.I e dashb d by tp

tlmwm

2

dctTaami

n the center, the photoelectron intensity is zero (node). The intensity profile on thy linearly polarized light. This distribution is the sum of three distributions causeolarized light. (d) The case of the p state excited by circularly polarized light.

he f wave [42]. In this case, the azimuthal quantum number is= 3 and the wave functions with magnetic quantum numbers= −l,−l + 1, . . . ,0, . . . , l are excited with equal probability. The

ave function of the Auger electron is of importance in the above-entioned reconstruction process.

.8. Scattering process

Theory of diffraction includes the kinematical theory and theynamical theory. In the former case, the multiple scattering pro-ess is neglected whereas the latter case, it is considered. Althoughhe dynamical theory is accurate, it has certain disadvantages.
he influence of multiple scattering depends on the scatteringmplitude. For example, when the scattering amplitude is 1%, themplitude of the doubly scattered wave is 0.01%. In this case, theultiple scattering effect is negligible. However, when the scatter-
ng amplitude is 50%, the amplitude of the doubly scattered wave

ed line is shown in the upper part of the image. (b) The case of the p state excitedhe three initial states, px , py , and pz . (c) The case of the s state excited by circularly

is 25%. In this case, the multiple scattering effect should be consid-ered. When the kinetic energy of an electron is low, the scatteringamplitude is large in all directions and the full multiple scatteringtheory must be considered. For such cases, a theory based on theGreen function involving the tracing of the multiple-scattering pathwas developed [43,44]. On the other hand, an electron with a higherkinetic energy can be treated by using the dynamical theory with-out the back-scattering effect. Here, we describe our theory basedon partial wave expansion.

2.9. Partial wave expansion

In partial wave expansion, the incident electron wave of the scat-terer is expanded to the partial spherical waves centered at thescatterer. The scattered wave is obtained by applying the phaseshift for each partial spherical wave. Here, a scatterer atom locatedon a is targeted. The incident wave at the scatterer is written as

2 opy a

ϕw

ϕ

w

ϕ

owt

otwT

wca

ϕ

t

|

etr

e

g

ϕ

jraas

f

�t

ϕ

wbt

e


ˆ L(k, r, �,�). Within the framework of the kinematical theory, thisave is approximated by the emitted wave from the emitter as

ˆ L(k, r, �,�) = ϕL(k, r, �,�). (27)

In the theory of diffraction, we consider the sum of the scatteredaves from another scatterer as

ˆ L(k, r, �,�) = ϕL(k, r, �,�) +∑j

L(k, r, �,�,aj). (28)

In this case, without the determining the scattered wave fromther scatterers, the incident wave is not fixed and the scatteredave is not calculated. Therefore, the dynamical theory is difficult

o handle.We describe the kinematical theory for the simplest case. In

rder to calculate the scattering, the coordinate origin should be seto the location of the target scatterer. Here, this coordinate system isritten as r′, i.e. r′ = r − a. The incident wave is written as ϕL(k, r′).

he incident wave expanded partial wave is written as ϕL(k, r′,a).Here, a rough approximation is introduced. An excited electron

ave is approximated by a simple spherical wave Eq. (3). At theoordinate system of the target scatterer, the wave function is givens

ˆ s(k, r′,a) = 1√4�

eik|r′+a|

|r′ + a| . (29)

Then, an additional approximation that the spherical wave nearhe scatterer is approximated by a plane wave is introduced:

limr′ |→0

ϕs(k, r′) ≈ 1√4�

eika

aeikr′

. (30)

ika/adescribes the change in phase and amplitude after an electronravels distance a. When an atom is located on the z-axis, Rayleigh’sule:

ikz =∞∑l=0

(2l + 1)iljl(kr)Pl(cos �) (31)

ives the partial wave function

ˆ (k, r′,a) = 1√4�

eika

a

∞∑l=0

(2l + 1)iljl(kr′)Pl(cos �′). (32)

l(x), Pl(cos �) are spherical Bessel function and Legendre function,espectively. The partial wave for the plane wave can be derivednalytically. However, this approximation is rough for scatterertoms located near the emitter, since the difference between thepherical wave and the plane wave is quite large.

On the other hand, in the case of the simple s wave, there is aormula that gives the partial wave directly.

eik|x−x′ |

|x − x′| = ik∞∑l=0

(2l + 1)jl(kx)h(1)l

(kx′)Pl(cos�)(x′ > x). (33)

is the angle between x and x′. When x = r′ and x′ = −a are used,he partial wave function is given as

¯ L(k, r′,a) = ik√4�

∞∑l=0

(2l + 1)h(1)l

(ka)jl(kr′)Pl(cos�). (34)

This approximation gives better results than that of the plane
ave. However, this function cannot treat the wave function caused
y the dipole transition, the Auger transition, or the multiple scat-ering.

Therefore, a formula that gives the partial wave for a generallectron wave is required. When the coefficient for the partial wave


is written as CLlm(a), the general equation can be written as

ϕL(k, r′, �′, �′,a) = k∑lm

CLlm(a)il+1jl(kr′)Ylm(�′, �′). (35)

The coefficient CLlm(a) is given by using the following integral.

CLlm(a) = (−i)l+1

∫ ∫ϕL(k, R′, �′, �′)Y∗

lm(�′, �′) sin �′d�′d�′

jl(kR′). (36)

This integral should be solved with a high degree of accuracybecause the spherical harmonic is a type of oscillating function. Inaddition, when jl(kR′) is zero, CLlm(a) cannot be obtained. Therefore,the parameter R′ should be set such that jl(kR′) has a finite value.

Here, we introduce a calculation technique. If this integral calcu-lations is solved using different values ofR′ for each l, the calculationrequires a long time because values of ϕL(k, R′, �′, �′) for differentvalues of R′ need to be estimated. When a particular value of R′ isdetermined, it is only required for the estimation of ϕL(k, R′, �′, �′)at that value. In order to find the particular value of R′ that givesfinite vales of jl(kR′) in l = 0∼lmax, the following function is effective.

F(kr) =∣∣∣∣∣lmax∏l=0

jl(kr)

∣∣∣∣∣ . (37)

The value of R′ that gives the maximum of this function shouldbe adopted.

Images of the partial wave expansion are shown in Fig. 14. Theemitted wave is set to be a spiral wave. The wave inside the circles isreconstructed by the partial waves. The radius of the reconstructedwave increases with the angular momentum l.

2.10. Potential of atoms

The scattered wave function is given by applying the phase shiftsto the incident wave ϕL(k, r′,a) expanded into the partial waves. Thephase shifts are given by solving the Schrödinger equation using ascattering potential. This potential consists of the Coulomb poten-tial of the atomic nucleus, that of other electrons, and the exchangeinteraction. This potential can be approximated by using the densityfunctional formalism

V(r) = Ven(r) + Vee(r) + Vex(r), (38)

Ven(r) = −Z|r| , (39)

Vee(r) =∫

(r′)|r − r′| dr′, (40)

Vxc(r) = −3˛(

38�(r)) 1

3. (41)

Z and (r) denote the charge of the atomic nucleus and the elec-tron density, respectively. It is known that a discrete variational X�(DV-X�) calculation, the calculated results agree with the exper-imental results when the parameter ˛ is set to ˛ = 0.7. In thecase of a photoelectron having an energy of several hundred eV,the effective potential mainly consists of the spherically symmet-ric attracting force around the scatterer atomic nucleus. Therefore,the muffin-tin approximation is effective. The radial function (r)of the electron density in the muffin-tin sphere is required. (r)can be easily obtained by the Thomas–Fermi approximation. How-ever, this is a rough approximation, and it gives more a broadened
electron density. In the calculation of the scattering, this approxi-mation makes the forward focusing peak stronger than it actuallyis. A better approximation method is the use of the radial functionsfor the inner shell electrons. The atomic potential can be obtainedby using the Slater functions that describe the radial function of


F citedi ntumse

tc

2

atftT

j

btos

ısio

ig. 14. (a) Real part image of the original wave function, which is a photoelectron exs 400 eV. (b) The result of a partial wave expansion calculation. The angular momexpansion using l = 0–10. (d) Partial wave expansion using l = 0–4.

he inner shell electrons. In this paper, the muffin-tin potential isalculated by using this approximation.

.11. Phase shift calculation

The effective radius of the muffin-tin potential is written as b,nd the radial wave function inside a is written as Rin

l(r). The func-

ion Rinl

(r) is given by numerically solving the Schrödinger equationrom the origin to the muffin-tin edge. Beyond the muffin-tin poten-ial is free space, and the incident wave function is given by Eq. (35).he Bessel function can be written as

l(kr′) = h(1)(kr′) + h(2)(kr′)

2. (42)

This equation implies that the inward and outward waves arealanced in the same amplitude and phase. In the scattered status,he amplitudes of two waves must be the same; however, the phasef the outward wave is shifted. Here, the wave function L for thecattered status applying the phase shift to Eq. (35) is given as

ˆL = k

∑lm

CLlm(a)il+1 (e2iıl h(1)l

(kr′) + h(2)l

(kr′))2

Ylm(�′, �′). (43)

l is the phase shift. In order to determine the phase shift, themooth connection condition between the radial function of thenside wave function Rin

l(r′) of the muffin-tin sphere and that of the

utside wave function Routl

(r′) is used. The radial function of the

by circularly polarized light from the s state. The kinetic energy of the photoelectronl = 0–19 were used. The waveform inside the circle is reproduced. (c) Partial wave

outsize wave is given as

Routl (r′) = (e2iıl h(1)l

(kr′) + h(2)l

(kr′)). (44)

The following equation is used for the smooth connection.

ddr′ R

outl

(b)

Routl

(b)=

ddr′ R

inl

(b)

Rinl

(b). (45)

Therefore, the phase shift ıl is given by

e2iıl = −˛h(2)l

(kb) − ddr′ h

(2)l

(kb)

˛h(1)l

(kb) − ddr′ h

(1)l

(kb), (46)

˛ ≡ddr′ R

inl

(b)

Rinl

(b). (47)

Since the phase shift is defined as 2iıl , the effective range is0 ≤ ıl < �. The profile of the phase shift ıl for C, Si, and Cu is shownin Fig. 15. The phase shift ıl of a large values of l increases withthe kinetic energy. In addition, the phase shift ıl increases withincreasing atomic number.

2.12. Scattered wave function

The wave function involves both the incident and scatteredwaves. Therefore, the scattered wave is given by removing the inci-

204 T. Matsushita et al. / Journal of Electron Spectroscopy a

FT

d

ts

L L

|ah|<|ac |L h

The multiple scattering calculation can be solved when the par-

ig. 15. The energy dependence of the phase shift. (a) The case of a carbon atom. (b)he case of a silicon atom. (c) The case of a copper atom.

ent wave from ( − ϕ)

L(k, r′, �′, �′) = k∑lm

CLlm(a)il+1 (e2iıl − 1)2

h(1)(kr′)Ylm(�′, �′).

(48)

Fig. 16 shows the real space image of the wave function whenhe excited electron wave is an s wave with Ek = 400 eV, and it iscattered by the Cu atom. (a) and (c) show the real part of the wave


function. (b) and (d) show the probability density distribution. Astrong forward focusing peak appears on the extended line con-necting the emitter atom and the scatterer atom. An interferencepattern appears between the emitter and the scatterer in the prob-ability density distribution, and it indicates interference betweenthe incident wave and the scattered wave.

This wave function is observed at a great distance. Eq. (48)describes the scattered wave function, the origin of which is thescatterer. This relation ship is used in order to return to the coor-dinate system in which the origin is the emitter atom. Here, thefollowing equation is used.

limr→∞

kr′ = limr→∞

k√

|r − a|2 = kr − k · a. (49)

The wave function at a great distance is already defined by Eq.(6), and it is given by

L(k,a) = e−ik·a∑lm

CLlm(a)(e2iıl − 1)

2Ylm(�,�). (50)

The amplitude of the scattered wave function | s(k,a)|2 isshown in Fig. 17(a) and (b). Here, the incident wave is not con-sidered.

The amplitude decreases when the atomic distance betweenthe emitter and the scatterer atom increases. The amplitude inthe forward scattering region is quite large, and it decreases atapproximately 25◦. This feature is of great importance in the above-mentioned multiple-scattering calculation.

Fig. 17(c) and (d) shows the function |ϕs(k) + s(k,a)|2; thisfunction involves both the incident and scattered wave functions.The function is oscillated, and it reveals the interference withthe incident wave. The amplitude of the scattered wave is smalloutside the forward scattering region (> 25◦); however, a stronginterference appears. Fig. 18 shows the energy dependence of thefunction |ϕs(k) + s(k,a)|2. The scatterer is a Cu atom located ataz = 0.255 nm. For Ek = 100 and 200 eV, these functions have acomplex structure. The function of Ek ≥ 400 eV becomes a type ofthe simple oscillating function. This feature has an advantage withregard to the above-mentioned reconstruction calculation. There-fore, it is preferable to measure an electron hologram using theelectron with Ek ≥ 400 eV.

2.13. Multiple scattering

The single-scattering calculation is given by the methoddescribed previously. However, in the case of electron diffrac-tion, multiple scattering must be considered because the scatteringamplitude is large. When an kinetic energy of the electron is greaterthan 400 eV, the scattering amplitude is large only in the forwardscattering region, as shown in Fig. 17. Therefore, it is possible toneglect the multiple scattering of the back-scattering. Here, a tar-get scatter atom located at ac is considered. The scattered wavesfrom other scatterer atoms located on r ≥ |ac | are the back-scatteredwaves. Therefore, these waves can be neglected since their ampli-tude is small. The incident wave of the target scatterer is givenby

ϕ (k, r) = ϕ (k, r) +∑

(k, r,a ). (51)

tial wave expansion calculation described above is applied to thisincident wave function. In this method, one can obtain the scat-tered waves in ascending order of the atomic distance between theemitter and the scatterer.


F ited wT locatef istrib

tooMp(si

2

lFtbiwttitat[

2

me

2

t

ig. 16. The real space image of the wave function scattered by a Cu atom. The exche real part of the wave function. The Cu atom (scatterer) indicated by the circle isunction. The Cu atom (scatterer) is located at 0.51 nm. (d) The probability density d

The effect of the multiple scattering effect is significant whenhe scatterer atoms are located on a line. Fig. 19 shows the resultsf the single- and multiple-scattering calculations. The structuresf the forward focusing peaks are found to be remarkably different.ultiple-scattering has the effect of making the forward focusing

eak smaller. The interference patterns in the back scattering regionx< 0 nm) in Fig. 19(b) and (d) resemble each other. The multiple-cattering effect in the back scattering region is smaller than thatn the forward scattering region.

.14. Spiral wave excited by circularly polarized light

In the case of a photoelectron excited by circularly polarizedight, the emitted wave function can be described as a spiral wave.ig. 20 shows the results in this case. A Cu atom is used as a scat-erer, and the incident angle of the circularly polarized light is set toe perpendicular to the paper. The initial state of the photoelectron

s set to be the s, p(m = 1), d(m = 2), f (m = 3) states. The excitedave is a spiral wave, as shown in the figure, and the structures of

he spiral waves differ from each other. This figure indicates thathe position of the forward focusing peak caused by this scatterers shifted from the extended line between the emitter and the scat-erer. The amount of shift increases in order for s,p(m = 1),d(m = 2),nd f (m = 3). This is because in the case of a spiral wave, the elec-ron does not move linearly near the emitter atom (Daimon effect)41,45].

.15. Simulation of electron hologram

In order to simulate the electron hologram, the effects of theean free path and refraction at the surface should also be consid-

red.

.16. Mean free path

The theory mentioned above is the elastic scattering process ofhe electron. In the solid, an inelastic scattering process also occurs,

ave is set to be an s wave with Ek = 400 eV. The emitter is located at the origin. (a)d at 0.255 nm. (b) The probability density distribution. (c) The real part of the waveution.

and it decreases the intensity of the electron. The decay length is themean free path. The mean free path depends on the kinetic energy ofthe electron [46]. In the energy region used in electron holography,the mean free path becomes longer with increasing kinetic energy.The length of the mean free path is of the order of a few nanometers.Therefore, only those electrons that are excited near the surface canbe emitted from the surface. In addition, when the angle from thesurface normal is increased, the traveling distance becomes longer,as shown in Fig. 21. The intensity from the atom with depth d fromthe surface is given as

I = I0 exp( −d cos �

), (52)

where � and are the angle between the surface normal and themotion direction of the electron and the mean free path, respec-tively. When the angle � is increased, only those electrons that arelocated near the surface can be emitted from the surface. In the sim-ulation of the electron hologram, it is necessary to determine thecluster size and shape by considering the effect of the mean freepath.

2.17. Refraction effect at the surface

When an electron moves from the surface to vacuum, refrac-tion occurs because of the inner potential, which is the differencebetween the energy origin of the solid and the vacuum level. Aschematic view is shown in Fig. 22. In order to obtain the refractedangle, the law of conservation of momentum is used for the hor-izontal component of the momentum of the electron. The wavenumber in a solid is given by

h2k2= Ek. (53)

2m

The wave number in the vacuum is given by

h2k2

2m= Ek = Ek − V0. (54)


Fig. 17. (a and b) The scattered wave amplitude | s(k,a)|2 caused by a Cu atom. Theposition of the Cu atom is indicated in the figure. The kinetic energy of the electronis

k√�ahsta

s 400 eV. (c and d) The interference pattern between the incident wave (s wave) andcattered wave caused by a Cu atom (|ϕs(k) + s(k,a)|2).

The refracted angle is given by

ˆ sin � = k sin �, (55)

Ek sin � =√Ek sin �. (56)

and � are the angles of the motion direction of the electron in
solid and in vacuum, respectively. In order to simulate electronolography or process the data, the refraction effect must be con-idered. For example, for an electron with Ek = 200 eV and � = 70◦,he angle � is approximately 80–90◦. For an electron Ek = 1000 eVnd � = 70◦, the angle � is approximately 71–72◦. The refraction
Fig. 18. Energy dependence of |ϕs(k) + s(k,a)|2. The scatterer is a Cu atom locatedat az = 0.255 nm.

effect becomes negligible with an increase in the kinetic energy ofthe electron.

2.18. Simulated electron holograms

Photoelectron holograms and an Auger electron hologram aresimulated using the theory described above. In order to directlycompare the experimental hologram shown in Fig. 3, the sphericalcluster of the Si atoms with 2.0 nm radius and the s wave electronwith Ek = 1355 eV are used in a simulation. The results are shownin Fig. 23. The cluster size was determined by considering the meanfree path of the electron. The decay effect and the refraction effectwere not applied to the calculation. The angular resolution was setto be 2◦.

In addition, the simulation for the experimental hologramshown in Fig. 8 was carried out using a spherical cluster of the Siatoms with 1.5 nm radius. It is shown in (Fig. 24). The initial state ofthe photoelectron was set to the 2p state. The excitation light wasset to be circularly polarized light, and the incident angle was setto be normal to the paper. The kinetic energy of the excited photo-electron was set to be Ek = 820 eV. The angular resolution was setto be 3◦.

These simulated results exhibit a good agreement with theexperimental results, and this fact implies the effectiveness of thetheory described above.

Fig. 25 shows experimental and simulated L3VV Auger electronholograms of Cu(0 0 1). The kinetic energy was set to be Ek = 914 eV.The emitted wave functions for the simulation were set to be the s,p, d, f, or g waves. The simulation using the f wave exhibited a goodagreement with the experimental result [29,42].

3. Atomic image reconstruction algorithm

Fourier transform was used in previous atomic image recon-struction methods. These algorithms are based on the theoreticalspeculation mentioned below. First, the excited wave function isapproximated by the s wave. In addition, the interference termbetween the scattered waves is neglected because the amplitudes


F idereda c) TheT

oT

�

fa

f

af

�

vt

U

i[cai

U

fsHasa

ig. 19. (a) The real part of the wave function. The multiple scattering effect is consre located at 0.255 nm and 0.51 nm. (b) The probability density distribution of (a). (he probability density distribution of (c).

f the scattered waves are smaller than that of the incident wave.he hologram function (Eqs. (50) and (12)) can be approximated by

(k) �∑h

2Re[feikah−ikah

ah

]. (57)

denotes the atomic scattering factor, which is given form Eq. (50)s

=∑lm

ae−ikaCLlm(a)e2iıl − 1

2Ylm(�,�). (58)

Then, an additional rough approximation is introduced. Thetomic scattering factor f is approximated as f∼1. The hologramunction is given by

(k) �∑h

2Re[eikah−ikah

ah

]. (59)

This equation is a type of Fourier series of the atomic positionector. Therefore, when the Fourier inverse transform is applied tohe hologram function

(r) = |∫�(k)e−ikr+ikr dk|2, (60)

t was considered that the atomic image can be reconstructed15,16]. However, the atomic scattering factor f is a function of aomplex variable. The approximation f∼1 was found to be unsuit-ble. Then, algorithms using the functionW(r,k) that corrects thenfluence of the function f were proposed.

(r) = |∫W(r,k)�(k)e−ikr+ikr dk|2. (61)

SWIFT using W = f −1 [17,18], SWEEP using W as the windowunction [19–23], differential holography usingW = ∂/∂Ek [25], and
o on were among some of the algorithms that were proposed.owever, it was reported that it is difficult to stably reconstruct antomic image by using these algorithms [24]. The strong forward-cattering peak and multiple scattering effects can induce imageberrations. In addition, the effect of the excited wave function also
. The emitted wave is set to be an s wave and Ek = 400 eV. Two Cu scatterer atomsreal part of the wave function. The multiple scattering effect is not considered. (d)

causes image aberrations. On the other hand, these algorithms arebased on the multi-energy hologram. Therefore, problem of the longmeasurement time must also be considered.

We have proposed other algorithms that do not use the Fouriertransform. We introduced the scattering process of the electron andthe wave function of the emitted wave to the algorithms. Therefore,the three-dimensional atomic image can be reconstructed from asingle-energy hologram. In these algorithms, the scattering patternfunction plays an important role. Next, we describe the scatteringpattern function.

3.1. Scattering pattern function

The amplitude of the scattered wave function is smaller thanthat of the incident wave. Therefore, the interference term in Eq.(12) is negligible.

�(k) �∑L

∑h

[2Re[ϕ∗

L(k) L(k,ah)] +∣∣ L(k,ah)

∣∣2] . (62)

This function has a simple structure. Here, we define the scat-tering pattern function

t(k,a) ≡ |a|∑L

[2Re[ϕ∗

L(k) L(k,a)] +∣∣ L(k,a)

∣∣2] . (63)

Here, the coefficient |a| is included in order to correct that theweakening of the amplitude of the scattered wave with increasingatomic distance between the emitter and the scatterer. This coef-ficient is of importance in the reconstruction algorithm. Using thisfunction, the hologram function can be written as

�(k) =∑h

t(k,ah)|ah|

. (64)

The hologram function can be described as the sum of the
scattering pattern functions. This scattering pattern function is asix-dimensional function, and it depends on the excitation processof the photoelectron and Auger electron.
The scattering pattern functions caused by the emitted electronwith the s wave and a kinetic energy of 1000 eV and the Si scat-


Fig. 20. (a) Real part of the wave function excited by the circularly polarized light from the s (l = 0, m = 0) initial state. The Cu scatterer is located at x = 0.255 nm. (b) Theprobability density distribution of (a). (b) That in the case of the p (l = 1,m = 1) initial state. (d) The probability density distribution of (b). (e) In the case of the d (l = 2,m = 2)initial state. (f) The probability density distribution of (c). (g) That in the case of the f (l = 3,m = 3) initial state. (h) The probability density distribution of (c).

Fig. 21. Electron path length and the angle of the electron.Fig. 22. Schematic view of the refraction effect. An electron is refracted at the sur-face.


Fig. 23. (a) Simulated photoelectron hologram of Si(0 0 1). The parameters of the simulation are as follows. Spherical cluster radius: 2 nm. Emitted wave function: s wave.Kinetic energy: 1355 eV. Angular resolution: 2◦ . (b) Experimental hologram measured by using unpolarized light (see Fig. 3).

F e simI Angul

tcfiissafbsftfstpatawa

ig. 24. (a) Simulated photoelectron hologram of Si(0 0 1). This image shows I/I0. Thncident angle of circularly polarized light: surface normal. Kinetic energy: 820 eV.

erer are shown in Fig. 26. Fig. 26(a) shows the scattering patternaused by the scatterer located on the z-axis at z = 0.235 nm. Theorward focusing peak appears on the z-axis, and the ring-shapednterference pattern appears around it. Fig. 26(b) shows the scatter-ng pattern caused by the atom located on the opposite side of thathown in Fig. 26(a). The scattering pattern is also inverted. Fig. 26(c)hows the scattering pattern caused by the scatterer located onn inclined arrow. The center of the scattering pattern is turnedorward the arrow. Fig. 26(d) shows the scattering pattern causedy the scatterer located farther. The spatial frequency of the ring-haped interference pattern increases with the atomic distance. Inact, the center of the scattering pattern indicates the direction ofhe scatterer, and the spatial frequency of the ring-shaped inter-erence pattern indicates the atomic distance. In the case of thewave, the scattering pattern function has a cylindrical symme-

ry around the atomic position vector a. Therefore, the scatteringattern function can be written as a function t(k, a, cos �) of the
tomic distance a, the wave number k, and the angle �. Fig. 27 showshe function t(k, a, cos �). The spatial frequency increases with thetomic distance or the kinetic energy. The peak at cos �∼1 is the for-ard focusing peak. The first interference ring, which is the first ring
round the forward focusing peak indicated by the arrow. When the

ulation parameters are as follows. Spherical cluster radius: 1.5 nm. Initial state: 2p.ar resolution: 3◦ . (b) Experimental hologram (see Fig. 8).

atomic number increases, the first ring position shifts to a lowerangle. This shift is due to the increase in the phase shift caused bythe scattering potential, which becomes deeper with an increase inthe atomic number.

3.2. Nodes in scattering pattern function of Auger electron

In the case of an Auger electron, an electron is emitted isotropi-cally. The scattering pattern function of an Auger electron also hascylindrical symmetry around the atomic position vector a. How-ever, the wave function of the emitted electron is not always an swave. Fig. 28 shows the scattering pattern function with changingwave functions of the emitted electron. In the case of an s wave,there is no node in the scattering pattern. In the case of a p wave,there is a node (t(k, a, cos �) = 0) at � = 90◦. The sign of the scat-tering pattern for � > 90◦ is inverted. This type of node appears atPl(cos �) = 0, where l is the orbital quantum number of the emitted
electron.
This phenomenon can be described by using the scattererlocated on the z-axis. In the case of an Auger electron with orbitalnumber l, the wave functions of m = −1,−l + 1, . . . ,0, . . . , l areexcited. The wave function with m = 0 has a node at Pl(cos �) = 0,


F nergyp on is s

aawtatma

ig. 25. (a) An observed L3VV Auger electron hologram of Cu(0 0 1) with a kinetic e, d, f, and g angular momenta for a spherical cluster of 241 atoms. Angular resoluti

nd the interference disappears at the angle where Pl(cos �) = 0. Inddition, the sign of the scattering pattern is inverted in the regionhere Pl(cos �) < 0, because the sign of the reference wave is nega-

ive. The other wave functions withm /= 0 have a node at the z-axis,nd the amplitudes of the wave functions are very weak. A scat-ered wave is hardly formed. Therefore, the interference pattern

ainly originates from the wave function withm = 0, and the nodesppear in the scattering pattern function. In the case of a scatterer

of 914 eV. (b)–(f) Simulated multiple-scattering patterns from emitted waves of s,et to 3◦ .

not located on the z-axis, the same discussion can be applied whenthe new z-axis is set to be the position vector of the scatterer atom.

3.3. Atomic image reconstruction algorithms

Eq. (64) yields the hologram from the atomic arrangement. Inorder to obtain the atomic arrangement from the hologram, algo-rithm to resolve Eq. (64) is required. Some methods that use a


F emittea a scat( ted at

nna

mtdabaHarsga

3

ddattt

gwa

ig. 26. Scattering pattern function caused by a Si atom. The kinetic energy of thend S indicate the emitter atom and the scatterer atom, respectively. (a) The case forc) The case for a scatterer not located on the z-axis. (d) The case for a scatterer loca

onlinear equation treat the atomic position vector as a variableumber (a type of R-factor method); these methods usually requiregood initial atomic model.

We consider two algorithms to reconstruct the atomic arrange-ent without an initial atomic model. The first algorithm utilizes

he cylindrical symmetry of the scattering pattern function. One-imensional data are calculated from the hologram using anveraging procedure, and then, the atomic arrangement is obtainedy using a fitting calculation from the one-dimensional data. In thislgorithm, a small error is introduced in the averaging procedure.owever, the amount of the calculation is low, and therefore, thetomic arrangement can be calculated rapidly. The second algo-ithm uses an N × N × N real space voxel. This algorithm directlyolves the equation that translates from the real space to the holo-ram. Although many calculations are required, and a more exacttomic image is obtained.

.4. Scattering pattern extraction algorithm (SPEA)

First, we consider an algorithm that uses the one-dimensionalata transformed from the hologram. This algorithm uses the cylin-rical symmetry of the scattering pattern function around thetomic position vector. First, the target vector a is selected, andhen, a rotation operation that moves a to the z-axis is applied tohe hologram. Fig. 29 shows circles on the sphere that correspond
o the latitude when the z-axis is considered as the axis of the earth.
The one-dimensional data are obtained by averaging the holo-ram data along each circle. This one-dimensional function isritten as �. When the target vector a does not correspond to the

tomic position vector, the one-dimensional data �becomes almost

d electron is 1000 eV, and the emitted wave function is an s wave. In the figure, Eterer located at z = 0.235 nm. (b) The case for a scatterer located at z = −0.235 nm.z = 0.543 nm.

zero, as shown in Fig. 29(c). When the target vector a correspondsto the atomic vector, the scattering pattern function remains in theone-dimensional data �, as shown in Fig. 29(b). Therefore, this aver-aging procedure can extract the scattering pattern functions causedby the scatterers on the target vector a. The one-dimensional datacan be represented as

�(k) =∑h

t(k,ah)|ah|

. (65)

This equation is similar to Eq. (64), however, vectors ah that areparallel to the target vector a are considered. Then, we introducedthe parameter gj that indicates the existence of an atom located onvector aj , where j is an integer index for each position. Then, theequation is modified as

�(k) =∑j

gjt(k,aj). (66)

This equation is equivalent to Eq. (65) when the parameter isset to be gj = 1/aj or gj = 0 when the atom does exist or does not,respectively. Eq. (66) can be regarded as the linear conversion fromthe atomic existence to the hologram. In order to calculate g from �,a fitting calculation is effective. The value gj must not be negativebecause it reveals the atomic existence. Here, a steepest descentmethod with a non-negative condition of constraint is used. In thismethod, g is calculated by minimizing the mean square error

C = 1N

∑i

|�(ki) −∑j

gjt(ki,aj)|2

�2i

. (67)


F ve funC the fir

ipftnc

ig. 27. Scattering pattern matrix. (a) SPM for C at Ek = 400 eV and 1000 eV. The wau at Ek = 400 eV and 1000 eV. Arrows on the left side image shows the position of

is an integer index for the one-dimensional data. The calculationrocedure is as follows. (1) gj is initially to 0. (2) The gradient vector

or the mean square error is calculated. (3) gj is corrected by usinghe steepest descent method. The corrected values that becomeegative are set to 0. (4) Steps (2) and (3) are iterated until theonvergence of gj .

The gradient vector is given by

∂C

∂gj=∑i

−2t(ki,aj)

�2i

⎡⎣�(ki) −

∑j′gj′ t(ki,aj′ )

⎤⎦ . (68)

ction ϕ is set to be an s wave. (b) SPM for Si at Ek = 400 eV and 1000 eV. (c) SPM forst interference ring.

Therefore, the mean square error decreases when gj moves inthe direction of −∂C/∂gj . We introduced a parameter ˛ to tune theamount of correction, and the correction procedure is given by

g(n+1)j

= g(n)j

− ˛ ∂C

∂g(n)j

. (69)

n is an index for the iteration. The parameter ˛ is determined byusing the steepest descent method. A series of these proceduresgives the one-dimensional atomic existence gj on the target vectora. Then, the three-dimensional atomic image is obtained by scan-ning the target vector a in three-dimensional space. This atomic

T. Matsushita et al. / Journal of Electron Spectroscopy a

Fp

idt

rasFWtst

isc

sSaadtt

i j

t(ki,aj) forms a matrix called the scattering pattern matrix. Mostof the matrix elements are non-zero. The number of elements isapproximately 20,000 × 2003 = 1.6 × 1011 in the case of the con-

ig. 28. Scattering pattern matrix for Cu Auger electron. Ek = 914 eV. (a) s wave. (b)wave. (c) f wave.

mage indicates the atomic existence image, and not the electronensity image. We call this algorithm the scattering pattern extrac-ion algorithm (SPEA).

This algorithm can avoid the conjugate image problem in theeconstruction from the single-energy hologram, i.e. the appear-nce the virtual image of the true image located on the pointymmetric position. The reason for this can be explained by usingig. 26(a) and (b). These two images are symmetric about a point.

hen the Fourier analysis is applied to these two scattering pat-erns, the two obtained values are the same and they cannot beeparated from each other. Therefore, a virtual image located onhe point symmetric position appears.

On the other hand, in the SPEA, the scattering pattern functions used directly in the reconstruction calculation, and these twocattering pattern can be separated from each other. Therefore, theonjugate image problem can be avoided.

In order to examine the effectiveness of the SPEA, we recon-tructed the atomic images from the experimental holograms ofi(0 0 1) and Ge(0 0 1). Here, it should be noted that two equivalent
tomic sites are involved in the unit cell of the Si and Ge crystal,s shown in Fig. 30(a) and (b). These two atomic sites cannot beistinguished by using the binding energy of the inner shell elec-ron. Therefore, the sum of the two holograms is observed. Whenhe atomic image is reconstructed from the sum of the two holo-
nd Related Phenomena 178–179 (2010) 195–220 213

grams, the reconstructed image is a sum of two real space images,as shown in Fig. 30(c). The atomic images on the (1 1 0) and (1 0 0)planes reconstructed from the Si(0 0 1) hologram (Fig. 3) are shownin Fig. 31. The surface normal is along the vertical axis. We suc-ceeded in reconstructing as many as 29 atoms with good resolutionalong the radial axis. Fig. 32 shows the atomic image reconstructedfrom the photoelectron hologram of Ge(0 0 1) by using the radiationlight (Fig. 4). Fig. 32(a) and (b) shows the cross sections of the (1 1 0)and (1 0 0) planes, respectively. Fig. 32(c) shows the stereo imageof the atomic image. The three-dimensional atomic arrangement ofapproximately 25 atoms can be seen directly.

3.5. SPEA with maximum entropy method (SPEA-MEM)

The SPEA has a drawback. In the averaging procedure, the con-tributions from those atoms not located along the target directionstill remain, as shown in Fig. 29(c). These may create virtual images.In order to eliminate this drawback, we proposed the followingalgorithm. Eq. (64) describes the transformation from the atomicposition to the hologram. We extended the equation as

�(k) =∫g(a)t(k,a)da. (70)

Here, the three-dimensional function g(a) is introduced. Whenthe function g(a) is defined as

g(a) =∑h

ı(a − ah)a

, (71)

Eqs. (62) and (70) are equivalent. Therefore, |a|g(a) indicates thethree-dimensional atomic distribution function. In order to obtainthe function g(a), we used a real space voxel with an N × N × Nmesh. The number of voxels areN = N3. The step of the mesh shouldbe finer than the spatial resolution of the electron hologram, asdiscussed below. Here, the voxelg(aj) located on aj is defined, wherej is an integer index for the voxel. The intensity of the hologram�(ki)measured at wave vector ki is defined. i is an integer index. Eq. (70)can be modified as

�(ki) =N∑j

gjt(ki,aj) Vj. (72)

Vj represents the volume of j-th real space voxel. It is a simplelinear equation, as shown in Fig. 33. However, this equation usu-ally cannot be solved by the simple gradient method. When eachaxis of ±1 nm real space is divided equally with a 0.01 nm reso-lution, the number of voxel is 2003. The number of data pointsfor a single-energy hologram measured with 1◦ square resolutionis approximately 20,000. Therefore, the number of unknown vari-ables is much greater than the number of the known data. When thenumber of measured data is increased by using the multi-energyhologram, this equation may be solvable; however, an energy-tunable light source and a long measurement time are required.

We have found that the function g can be estimated by the max-imum entropy method. We first discuss the features of t(k ,a ).

dition described above. When a double-precision value is used foreach matrix element, a memory space of 1.2 TB is required. There-fore, a large amount of calculations is involved. Very fast calculationalgorithms for the linear equation and the generation of the matrixelements of the scattering pattern matrix are therefore required.


Fig. 29. Illustration of the averaging process. The dashed line shows the original hologram �(k, �,�). The filled region shows the inclined hologram after the rotation process.The circles for the averaging process are indicated on the sphere. (b) The scattering pattern of an atom located on the z-axis. The graph on the right-hand side shows �(k, �p)calculated in the averaging process. (c) The scattering pattern and �(k, �p) of an atom that is not located on the z-axis.

Fig. 30. The atomic position from two equivalent emitters. (a) The atomic positions from emitter A. (b) The atomic positions from emitter B. (c) The atomic positions fromthe emitters A and B.


Fig. 31. The real space image reconstructed from the photoelectron hologram ofFig. 3. (a and b) The vertical slices of the (1 1 0) and (1 0 0) planes, respectively. Thevrcs

3

fvp

t

wdd

ertical axis is the direction of the surface normal. The brightness of the image cor-esponds to the parameter |a|g(a). The expected atomic positions are indicated byircles. The dashed-line circles indicate the atomic position from emitter A. Theolid-line circles indicate the atomic position from emitter B.

.6. Fast generation algorithm for scattering pattern matrix

In the case of an Auger emission or s wave, the scattering patternunction has a cylindrical symmetry around the atomic positionector a. By using this symmetry, the dimension of the scatteringattern function can be reduced.

(k,a) = t(k, a, cos �), (73)

here � is the angle between a and k. When a table of the three-imensional function t(k, a, cos �) is calculated in advance, the six-imensional function can be quickly generated from this table.

Fig. 33. Schematic view of the translation o

Fig. 32. Real space images reconstructed from the Ge(0 0 1) hologram (Fig. 4). Thevertical slices of the (1 1 0) and (1 0 0) planes are shown in (a) and (b), respectively.(c) A stereo image of the reconstructed three-dimensional real space.

In the case of circularly or linearly polarized light, the applicationof the fast generation algorithm was difficult, since the scatteringpattern functions do not have a cylindrical symmetry. In the caseof circularly polarized light, the forward focusing peak is shifted

f a real space voxel to a k-space pixel.


F of thez nds tor e ato[ f wav

brti

3m

sii

ig. 34. Reconstructed real space images. (a and b) The vertical slices [(0 1 0) plane]-Axis is the direction of the surface normal. The brightness of the images correspoadius the represent expected positions of Cu atoms, and dashed circles represent th(0 0 1) plane] at various z-positions of the atomic image reconstructed by using the

y the Daimon effect [41]. This shifts the atomic position in theeconstructed image; however, the amount of the order of a fewens of picometers and it can be obtained [47]. Therefore, the atomicmage shift caused by the Daimon effect is correctable.

.7. Application of iterative scaling method of maximum entropyethod

In order to calculate the function g, it is necessary to con-ider its characteristic and the information theory. The function gs non-negative, and is zero in most places. Therefore, the max-mum entropy method is quite effective [48–51]. We succeeded

atomic image reconstructed by using the s wave and f wave functions, respectively.|a|g(a). The emitter position is indicated by a small solid circle. Circles with 0.1 nm

mic positions that are not reconstructed by the calculation. (c) The horizontal slicese function.

in reconstructing the atomic arrangement by using the iterative-scaling algorithm of the maximum-entropy method. We called thisalgorithm SPEA-MEM (scattering pattern extraction algorithm withmaximum entropy method). The entropy is defined as

S = −N∑g(n)j

lng(n)j

g(n−1)− C, (74)

j j

�cal(ki) =N∑j

g(n)jt(ki,aj) Vj, (75)


F in Figu using

�

C

wrbio

i

ig. 35. The atomic images reconstructed from a Cu photoelectron hologram shownsing a normal scattering pattern function (Eq. (63)). (b) The images reconstructed

err(ki) = �exp(ki) − �cal(ki), (76)

= 1N

N∑i

∣∣�err(ki)∣∣2�2i

− 1, (77)

here n is an index for the iteration. �exp(ki), �cal(ki), and �err(ki)epresent the measured hologram, calculated hologram, and erroretween the measured and calculated holograms, respectively. �i

s the standard deviation of the noise. The entropy is maximized tobtain the real space voxel. In order to maximize the entropy

∂S

∂g(n)j

= 0, (78)

s used. The equation is modified as

∂S

∂g(n)j

= − lng(n)j

g(n−1)j

− 1 − ∂C

∂g(n)j

= 0. (79)

. 9. Expected atomic positions are indicated by circles. (a) The image reconstructedthe scattering pattern function without factor |a| (Eq. (83)).

Then,

g(n)j

= g(n−1)j

e

−1− ∂C

∂g(n)j (80)

is obtained. After g(n) is calculated using this equation, a scalingoperation is applied to g(n). This procedure is repeated until theconvergence of the functiong(n). This is the iterative scaling method.Appropriate values for the scaling parameter and the parameter are required. We introduced the following algorithm. Eq. (76) isredefined as

�err(ki) = �exp(ki) − (a�cal(ki) + b). (81)

Here, parameters a and b are introduced. These parameters are
optimized using the linear fitting method, which minimizes thefunction
∑i|�err(ki)|2. Parameters a and b represent the scaling

value and the background value of the hologram, respectively.Next, we describe the optimization of the parameter . This

parameter is important for the convergence performance. We used

2 opy and Related Phenomena 178–179 (2010) 195–220

t

Hem

gsiawtbhsrut

3f

pa|

t

iiataead

3

guoptdtˇmrlnttftit(tp

Here, we discuss the relationship between the spatial resolutionand the energy resolution of the electron. The electron analyzer hasan energy resolution E/dE = 10,000. For E/dE = 1000, the resolu-tion of the wavenumber of the electron is k/dk = 2000. According


he following equation:

= 1max(∂C/∂gj)

. (82)

It usually gives a tolerably good convergence condition.owever, it rarely falls into the oscillatory solution. Thevaluation algorithm of the parameter admits of improve-ent.

The images reconstructed from the Cu Auger electron holo-ram (Fig. 7) by using SPEA-MEM are shown in Fig. 34. Fig. 34(a)hows the results obtained by reconstruction using the scatter-ng pattern function caused by the s wave. Here, some artifactsppear. The wave function of Cu L3VV was considered to be an fave, and this affects the reconstructed image. Fig. 34(b) shows

he results obtained using the scattering pattern function causedy the f wave. The obtained result was better. Fig. 34(c) shows theorizontal slices on the each atomic layer. In the three-dimensionalpace, we succeeded in reconstructing as many as 102 atoms. Theseesults indicate that the atomic arrangement can be obtained bysing SPEA-MEM and the appropriate scattering pattern func-ion.

.8. Effect of normalization parameter of scattering patternunction

Here, it should be noted that the factor |a| in the scatteringattern function t(k,a) is important for the reconstruction of thetomic image. When the scattering pattern function without factora|

(k,a) =∑L

2Re[ϕ∗L(k) L(k,a)

]+ | L(k,a)|2, (83)

s adopted, the atomic image obtained using the hologram shownn Fig. 9 is shown in Fig. 35. The image is only clear near the emittertom, and the distant atomic image disappears. This is caused byhe asymmetry of the scattering pattern function t(k,a). Since themplitude of the function t(k,a) decreases with distance from themitter, the gradient of the mean square error ∂C/∂gj decreases fordistant atom. Therefore, gj for a distant atoms decreases, and theistant atomic image disappears.

.9. Spatial resolution of electron hologram

In order to evaluate the spatial resolution of the electron holo-ram, SPEA the polar coordinates are used. Here, we indicate it bysing the first nearest neighbor of the Si atom. The angular res-lution of the real space is related by the shape of the scatteringattern function. ˇ indicates the angle between the atomic posi-ion vector a and the axis of the averaging process v. The profile �epends on the angle ˇ, as shown in Fig. 36. When ˇ is increased,he profile � differs from the scattering pattern function. When= 5◦, peaks (a), (b), and (c) disappear. In this situation, infor-ation about the atomic distance is lost. Therefore, the angular

esolution is evaluated to be approximately 5◦. This is equiva-ent to approximately 0.02 nm at the position of the first nearesteighbor. The spatial resolution of the atomic distance betweenhe emitter and scatterer atoms is related to the frequency ofhe scattering pattern function. Fig. 37 shows a scattering patternunction caused by a scatterer located at 0.235 nm. The peaks ofhe scattering pattern function are indicated by (a)–(d). Peak (a)
s a forward focusing peak, which does not depend on informa-ion about the atomic distance. The locations of peaks (b), (c), andd) depend on the atomic distance. The effect of the atomic dis-ance increases in the order of peaks (b), (c), and (d); however, theeak intensity decreases in the order. When the angular resolu-
Fig. 36. �(�) for Ek = 1000 eV and a Si scatterer located at z = 0.235 nm. ˇ is definedas the angle between the axis of the averaging process and the scatterer positionvector, as shown in the illustration on the right-hand side.

tion of the electron analyzer is 1◦, the limitation of the estimationof the peak position is 1◦. The atomic distances are a = 0.224 nmand a = 0.248 nm when peak (c) is shifted at ±1◦. Therefore, theatomic resolution is approximately 0.013 nm, as shown in Fig. 37.However, in the case of reconstruction from experimental data, theatomic images are occasionally stretched because of the effect ofthe angular resolution and the contribution of the scattering pat-tern caused by other scatterer atoms. When the same discussionis applied to scatterer atoms located at various distance, a similarspatial resolution is obtained. Therefore, the spatial resolution forEk = 1000 eV is approximately 0.02 nm. For Ek = 400 eV, the spatialresolution is approximately 0.030 nm, and for 2000 eV, it is approx-imately 0.015 nm. The spatial resolution increases with the kineticenergy.

Fig. 37. Scattering pattern functions for Ek = 1000 eV and a Si scatterer located at0.224 nm, 0.235 nm, and 0.248 nm.

opy and Related Phenomena 178–179 (2010) 195–220 219

tiia

3

telaaedtalht[ce

g

R

T. Matsushita et al. / Journal of Electron Spectrosc

o this value, the spatial resolution of the atom located at 0.2 nms 0.1 pm. The spatial resolution caused by this energy resolutions negligibly small. Therefore, the energy resolution of the electronnalyzer does not affect the spatial resolution.

.10. SPEA-MEM with translational operation

We have proposed a reconstruction algorithm using a transla-ional operation. The above-mentioned algorithms only use thelectron hologram. When other information such as the trans-ational symmetry of the crystal is used in the reconstructionlgorithm, a clearer atomic image can be expected. Informationbout the translational symmetry can be measured by using low-nergy electron diffraction (LEED), reflection high-energy electroniffraction (RHEED), and X-ray diffraction. In addition, Kikuchi pat-erns that appear in the electron hologram also provide informationbout the translational symmetry. The information about the trans-ational symmetry does not indicate the atomic arrangement. Weave proposed a reconstruction algorithm that uses the transla-ional symmetry; this algorithm yields a clear atomic arrangement30]. This algorithm is based on SPEA-MEM. We have added a pro-edure that mixes the real space voxel g(n)

jwith that located at the

quivalent position in the neighboring unit cell. The mixing is

(r) =∑

|R|<Ra

|r + R||r| w(R)g(r + R), (84)

= la1 +ma2 + na3, (85)

Fig. 39. Atomic image reconstructed from the experimental single-energy PEH

Fig. 38. Schematic view of mixture of voxels by translational symmetry

where an is the unit vector of the crystal; Ra, the radius for themixing operation; w(R), a weight function. When w(R) = ı(R), the
effect of the translational mixing operation disappears. The effect ofthe translational mixing operation can be controlled by the carefulselection of w(R).
Here, we describe a case in which the effect of the transla-tional mixing operation is small. In this case, the weight function is

of Cu(0 0 1) using SPEA-MEM with the translational mixing operation.

2 opy a

g

w

wetstlmfTPbaitr

ft

4

dtAwsbwfctaitsiftmasuthe

A

cNnPeDcD

[

[[[[[

[

[

[[[

[

[[[

[

[[

[

[

[

[

[[

[[

[[[


iven by

(R) =(

1/C : |R| = 0v/C : |R| = |a|0 : others

, (86)

here v and C are the weight value and the normalization param-ter, respectively. The voxel value is only mixed with that ofhe neighboring unit cell. A schematic view of this procedure ishown in Fig. 38. In the case of a face-centered cubic (FCC) struc-ure, the voxel located at the (0 0 1) position mixes with thoseocated at (0 0 1 ± 1), (0 ±1,1), and (±1 0 1). However, it does not

ix with that located at (1/2 1/2 0). Even with such an imper-ect mixture, the atomic arrangement can be reconstructed clearly.he result of the application of the translational mixture to theEH of Cu is shown in Fig. 39. The shape of the atomic imageecomes spherical. Although there is no mixing between thetomic image of (0 0 1) and that of (1/2 1/2 0), each atomic images reproduced at its exact location. This result indicates that theranslational mixing method is quite effective in atomic imageeconstruction.

This fact also indicates the expandability of SPEA-MEM. In theuture, better atomic images will be reconstructed by using addi-ional information about the atomic arrangement.

. Conclusion

We have introduced theories for electron holography that wereeveloped by us. These proposed techniques enable the reconstruc-ion of three-dimensional atomic images from a photoelectron,uger electron, and inverse-mode electron holography. However,e are still developing and improving. For example, the recon-

tructed atomic image is clear in the forward-scattering regionecause the backscattering amplitude is smaller than that of the for-ard scattering. In the future, this will be improved. In addition, the

ast calculation algorithm for the scattering pattern is exact in thease of s wave and Auger electrons, but not in the case of photoelec-rons excited by polarized light. A new fast calculation algorithm islso required. These theories enable the bulk structure around anndividual atomic site or impurity determined by selecting the ini-ial state depending on the atomic species and chemical state. Byelecting the core level of an adsorbate, it is possible to investigatets surface structure. These theories will become extremely usefulor determining the positions of atomic nuclei around particulararget atomic sites. In addition, we have proposed a new measure-

ent method called diffraction spectroscopy [52]. The XAFS (X-raybsorption fine structure) and MCD (magnetic circular dichroism)pectra of each atomic layer of the surface can be measured bysing the photon energy and helicity dependence of the Auger elec-ron diffraction pattern (hologram). It will be possible to develop aighly advanced measurement method by use in combination withlectron holography.

cknowledgments

This work was performed with the approval of the Japan Syn-hrotron Radiation Research Institute (Proposal No. 2003B0088-M-np, 2004A0145-NM-np, 2004B0482-NM-np, 2005A0445-CM-p,2007A1278) and partly supported by Nano Technology Specially
romoted Research Projects from the Ministry of Education, Sci-nce, Sports, and Culture, Japan. The author deeply acknowledger. Yukako Kato and Dr. Fang Zhun Guo for many collaborativeontributions, and Dr. Takayuki Muro, Dr. Tetsuya Nakamura, andr. Toyohiko Kinoshita for their support and encouragement. The
[[

[


author also acknowledge Dr. Akane Agui and Dr. Akitaka Yoshigoefor the collaborative contributions at an early period of this work.

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