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Element Assignment for Three-Dimensional Atomic Imaging

by Photoelectron Holography

Tomohiro MATSUSHITA, Fumihiko MATSUI, Kentaro GOTO,Taku MATSUMOTO, and Hiroshi DAIMON

J. Phys. Soc. Jpn. 82 (2013) 114005

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Element Assignment for Three-Dimensional Atomic Imaging

by Photoelectron Holography

Tomohiro MATSUSHITA1�, Fumihiko MATSUI

2, Kentaro GOTO2,

Taku MATSUMOTO2, and Hiroshi DAIMON

2

1Japan Synchrotron Radiation Research Institute, SPring-8, Sayo, Hyogo 679-5198, Japan2Graduate School of Materials Science, Nara Institute of Science and Technology, Ikoma, Nara 630-0192, Japan

(Received May 29, 2013; accepted August 19, 2013; published online October 21, 2013)

Photoelectron holography has been expected to be a powerful tool for visualizing three-dimensional (3D) localatomic structures around a photoelectron emitter atom. Photoelectron emitter atom sites with different elements can bedistinguished by photoelectron kinetic energy. The elements of atoms reconstructed in atomic images, however, couldnot be assigned. We developed a method for the elemental characterization of the reconstructed atoms using a smalldifference of the scattered electron waves and demonstrated it using a measured InP(001) photoelectron hologram.Element assignment for both the emitter atom and the reconstructed atoms in 3D atomic images became possible.

KEYWORDS: photoelectron holography, photoelectron diffraction

1. Introduction

The principle of electron holography was proposed byGabor.1) In 1986, Szoke pointed out that a photoelectronemitter atom can act as an atomic-sized point source forelectron holography.2) The emission of photoelectronsspontaneously satisfies Gabor’s geometry of point sourceholography. A schematic view of the recording process forphotoelectron holography is shown in Fig. 1. Core levelphotoelectrons are excited by X-rays. Since the bindingenergy of core electrons is element-specific, photoelectronemitter atoms can be specified by the selective detection ofphotoelectrons with the corresponding kinetic energy. A partof the photoelectron wave is scattered by the surroundingatoms. An interference pattern of direct and scattered wavesappears in the photoelectron intensity angular distribution.As a result, this angular distribution can be regarded as ahologram that records three-dimensional (3D) atomic imagesaround the emitter atom. A hologram pattern caused by ascatterer atom has a strong forward focusing peak (FFP) andring patterns around it. The scatterer atom located fartheraway from the emitter atom forms finer ring patterns. TheFFP indicates the direction of the scatterer atom. Thefineness of ring patterns indicates the atomic distance. Thus,the photoelectron hologram provides a 3D image arounda photoelectron emitter atomic site without requiring anyinitial model or phase information. It is possible to assignnot only the bulk structure but also the surface structure andthe local impurity structure of the crystal. Therefore, it is apowerful tool for the analysis of 3D atomic structures.

Up to now, much effort has been devoted to the studyof atomic resolution holography using photoelectron,3–14)

Auger electron,15–18) reflection electron Kikuch-scatter-ing,19–22) and low-energy electron diffraction.23–25) Recently,photoelectron holograms excited by hard X-rays26,27) havebeen reported. In addition, internal-detector electron holo-graphy,28) which utilizes a kind of time reversal process ofphotoelectron holography, was developed. The contrastreverse pattern of photoelectron holograms was found inthe energy-loss electron angular distribution.29) The combi-nation of Auger electron diffraction and X-ray absorption

spectroscopy, which gives the spectra of each atomic layerat the surface, was reported.30) The 3D atomic imagereconstruction algorithm was also developed. The firstreconstruction algorithm based on Fourier transformationwas proposed by Barton.31,32) Atomic image reconstructionsusing an experimental electron hologram5,16) were alsoreported. The atomic image reconstructed using the Bartonalgorithm causes an atomic position shift33) because thisalgorithm ignores the strong phase shift effect and forwardfocusing effect caused by electron scattering. Manyresearchers exerted effort to eliminate the phase shift effectand forward focusing effect. Reconstruction algorithms,such as phase shift correction,34) the scattered-wave integral-transform method,35) scattered-wave-included Fourier trans-form (SWIFT),36,37) the small-window energy extensionprocess (SWEEP),3,4,38,39) the small-cone method,8,40) differ-ential holography,41) near-node holography42) and othermethods36,43,44) were proposed in previous studies. Most ofthese investigations focused on the improvement of theBarton algorithm to eliminate the phase shift and forwardfocusing effects. Len et al.45) have applied above thealgorithms8,31,32,37,40,43,44) to the study of the tungsten surface

(a)

Hologram

Emitter atom

Scatterer atom

Soft X-ray

Photoelectron wave

(b)Longer atomic distance

Forward focusing peak

Fig. 1. (a) Recording process for photoelectron holography. A

photoelectron excited by X-ray is scattered by the scatterer atom. The

interference of the scattered and direct waves forms ring-shaped interference

patterns, i.e., a photoelectron hologram. (b) When the atomic distance

between the emitter and scatterer atoms increases, the ring patterns become

finer.

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and bulk photoelectron holograms, and examined theirvalidity. The local structure surrounding the surface-emission sites was approximately determined by thesemethods; however, the reconstruction of bulk atomicarrangements was unsuccessful.

To solve the above problem, we have proposed ascattering pattern extraction algorithm using the maximumentropy method (SPEA-MEM),46–49) which was not based onFourier transform. This algorithm used the scattering patternfunction (SPF) as a basis function and the maximum entropymethod (MEM) for the fitting procedure.

In this study, we developed an advanced algorithm thatenables the reconstruction of 3D atomic images with theelement information of the scatterer atoms. Therefore, theelement analysis of both the emitter and scatterer atomsis realized. We estimated the effective conditions for theassignment of elements. We applied the algorithm using anexperimental photoelectron hologram of InP in order toshow its validity using real data.

2. SPEA-MEM

The SPEA-MEM is described as follows. The photoelec-tron hologram can be described as48,49)

�ðkÞ ¼Z

tðk; aÞgðaÞ da; ð1Þ

tðk; aÞ ¼ jajXL

2<½�LðkÞ¼�Lðk; aÞ� þ j¼Lðk; aÞj2; ð2Þ

where k is the wave vector of the photoelectron and tðk; aÞ isan SPF caused by the atom located at position a. The 3Datomic distribution function is given by jajgðaÞ. �LðkÞ is thewave function of the emitted photoelectron, where L is anindex for assigning the final excited state. ¼Lðk; aÞ is thescattered wave function caused by the atom located atposition a, which includes the phase shift effect and theforward focusing effect of the electron scattering process.

The atomic distribution function can be deduced bymaximizing the entropy S46,50) as

S ¼ �Xj

gðnÞðajÞ ln gðnÞðajÞgðn�1ÞðajÞ � �C; ð3Þ

C ¼Z j�errðkÞj2

�2ðkÞ dk; ð4Þ

�errðkÞ ¼ ��ðkÞ �Z

tðk; aÞgðaÞ da; ð5Þ

where a measured hologram is denoted by ��ðkÞ. ðnÞ indicatesthe number of iterations. � is a disposable constant to beevaluated. � is a standard deviation function of �. In order torepresent gðaÞ, a 3D mesh, voxel gða jÞ, where a j representsthe voxel position, is utilized. Since the SPF tðk; aÞ includesthe phase shift effect, the atomic position deviation issuppressed, unlike in the Barton algorithm.

The SPF is influenced by the atomic number of thescatterer atom because of the scattering potential. Weestimated the effect when the photoelectron hologram ofthe compound is analyzed by a single SPF for one kind ofelement. As discussed below, this effect is limited when thedifference between the atomic numbers is small.

When additional constraint conditions using prior infor-mation on the atomic structure such as the symmetry

introduced into the reconstruction calculation, the atomicimage becomes clearer. A translational symmetry providesa strong constraint condition.13,14,28,48,49) We have added aprocedure, which mixes a voxel gðaÞ with voxels gðaþ RÞlocated at the equivalent position in neighboring unit cells,into the SPEA-MEM (R is a translational vector). It givesclear 3D atomic images without artifacts even if thecompound is analyzed by the single SPF for one of theelements in the compound.

We tried to reconstruct a 3D element structure using ameasured photoelectron hologram. An InP(001) wafer wasselected as the sample. The experiment was carried out atBL25SU of SPring-8. The two-dimensional display-typeanalyzer that we developed51) was used. The angulardistribution of the core-level photoelectron intensities ofIn 3d with Ek ¼ 600 eV was measured, as shown inFig. 2(a). We applied the SPEA-MEM using the SPF forIn and the translational symmetry of the face-centered cubic.A voxel mesh with 0.01 nm resolution was utilized. Theresult is shown in Fig. 2(b). We succeeded in reconstructinga clear and exact 3D atomic structure of InP. Although theSPF for In was used, the atomic images of P at the sublatticesite were also reproduced at the exact position.

3. Theory of Element Assignment

As mentioned above, the SPF slightly depends on theelement of the scatterer atom. Here, we denote the SPF astZðk; aÞ, where Z represents the element. The atomic numberdependence of the SPF is shown in Fig. 3. The kineticenergy and wave function were set to 600 eV and the s wave,respectively. The SPFs exhibit an FFP and first-, second-,third-, . . . order interference peaks. These interference peaksmove to smaller angles with increasing atomic distance a,as shown in Fig. 1. Increasing the atomic number has asimilar effect on the interference peak positions as thedistance changes, but the effect is much smaller, as shown inFig. 3. This causes a small atomic position shift. This smallshift was suppressed in Fig. 2(b) because of the use oftranslational symmetry.

By using the above difference, the element analysis of thescatterer atom becomes possible. The calculation methodis described as follows. To begin with, an exact atomicstructure is required for the element assignment. Here, wedetermined the exact atomic positions using SPEA-MEMwith translational symmetry operation. Then, we estimatedall atom positions ai, and redefined the voxel gZðaiÞ that

Emitter (In)

(a) (b)

(100)

(010)

(001)

Unit cell

Ek=600eV

In 3d

Fig. 2. (a) Measured photoelectron hologram of InP(001). (b) Atomic

images reconstructed using SPEA-MEM.

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describes the occupation of the element Z at the atomposition ai. Therefore, the hologram function can bemodified as

�ðkÞ ¼XZ

Xi

tZðk; aiÞgZðaiÞ: ð6Þ

The candidate element can be determined using thephotoelectron spectrum. This equation can also be solvedusing Eq. (3). The translational symmetry operation was notapplied in the element reconstruction stage.

We applied the above improvement to the InP hologram.The result is shown in Fig. 4. The atomic images are coloredaccording to the ratio of occupation: ðgIn � gPÞ=ðgIn þ gPÞ.The ratio of occupation for the first-, second-, third-, andfifth-nearest neighbors were derived correctly. Those for thefourth-nearest neighbors were unclear. Element assignmentfor atoms beyond the sixth atom was inaccurate in thisresult.

The key point in the assignment of the element is the useof the voxels gZðaiÞ located at fixed atomic positions. Forexample, in the case of InP, when the analysis functionof In is applied to a hologram caused by a P atom, thereconstructed atomic image is shifted from the exact atomicposition along the radial direction because the interferencepeaks of the In analysis function are located at anglessmaller than that of the hologram caused by the P atom.

Therefore, when the voxel positions are fixed, the intensityof the voxels derived from the correct element is higher.Therefore, the element can be assigned.

In order to confirm the reasoning behind the assignment ofelements, we defined an evaluation function for the atomicimage as

FZ1;Z2ða; rÞ ¼

ZtZ1

ðk; aÞtZ2ðk; rÞ dk; ð7Þ

where tZ1ðk; aÞ and tZ2

ðk; rÞ describe a hologram functionand an analysis function, respectively. This equation givesthe radial atomic image of the scatterer atom of atomicnumber Z1, which has been analyzed with the analysisfunction of atomic number Z2. Here, we define the analysisfunction as

tZðk; aÞ ¼ tZðk; aÞwðkÞ; ð8Þwhere wðkÞ is a Hann window function that weakens thecontribution of the backscattering in order to improve theatomic image.

We estimated the evaluation function of InP. The resultsfor Z1 ¼ 49 (In) are shown in Fig. 5. Figures 5(a) and 5(b)show the results for Z2 ¼ 49 (In) and Z2 ¼ 15 (P),respectively. A direct comparison of the function F at a ¼0:3 nm is shown in Fig. 5(c). When the element of thescatterer atom and the analysis function are the same, theatomic image (peak) appears at a certain position (r ¼0:3 nm). The peak width is estimated to be approximately0.022 nm, which is related to the radial resolution of theatomic image. As shown in Fig. 5(c), when the hologramcaused by In is analyzed by the SPF of P, the peak is

t Z(

k, a

)

Z493625159

a = 0.3 nmE

k = 600 eV

s wave

FFP

0 30 60 90 120 150 180

InP

1st 2nd 3rd

θ (deg)

Fig. 3. Atomic number dependence of scattering pattern functions (SPFs)

tZðk; aÞ, where � is the angle between a and k.

0 25%-25%

(gIn-g

P)/(g

In+g

P)

InP

Emitter(In)

3rd(P)

4th(In)

4th(In)

5th(P)

6th(In)

1st(P)

2nd(In)

3rd(P)

7th(P)

(001)

(100)

6th(In)

8th(In)

9th(P)

10th(In)

Fig. 4. (Color online) Reconstructed atomic images colored according to

ratio of occupation ðgIn � gPÞ=ðgIn þ gPÞ.

0.500.450.400.350.300.250.20r (nm)

0.50nm

0.45

0.40

0.35

0.30

0.25

0.20

0.50nm0.400.300.20a (nm)

0.50nm

0.45

0.40

0.35

0.30

0.25

0.20

0.50nm0.400.300.20a (nm)

(a) (b)

Z2=In

Z2=P

(c)

Z1=InZ

1=In Z

2=In Z

2=P

0.044 nm

FZ

1,Z

2(a =

0.3

nm, r

)

0.022 nm

r (n

m)

r (n

m)

Fig. 5. Evaluation function FZ1;Z2ða; rÞ. The atomic number of the

hologram (Z1) is set to be 49 for In. The final state is set to be the s

wave and its kinetic energy is 600 eV. (a) and (b) show the results using the

evaluation functions for Z2 ¼ 49 and 15 for In and P, respectively.

(c) Evaluation function FZ1;Z2ða; rÞ at a ¼ 0:3 nm.

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asymmetric and the position is shifted to be approximately0.044 nm further away. As shown in Fig. 5(b), this valueremains the same for all atomic positions. From theseresults, element analysis is made possible when the atomicimage shift caused by the atomic number difference is largerthan the radial resolution of the atomic image. In the case ofInP, this condition is satisfied.

This method requires that the position shift caused by theatomic number difference is larger than the spatial resolutionof holography. Finally, we evaluated the peak shift for allelements at Ek ¼ 600 eV, and the results are shown inFig. 6. Z1 and Z2 show the atomic numbers for the hologramfunction and analysis function, respectively. When theatomic number (Z2) of the analysis function is smaller thanthat of the hologram function (Z1), the atomic image shiftsfurther away. When the absolute value of the shift is largerthan the spatial resolution, the assignment of the scattereratom element is possible. Therefore, the region over 0.022nm can resolve the element. In the figure, the Z1 ¼ 2Z2 andZ2 ¼ 2Z1 lines have also been shown. In general, we canconclude that when the atomic number difference is morethan twofold, this method is effective.

As mentioned above, the exact atomic positions aroundthe emitter are required for the above element assignment.SPEA-MEM with translational symmetry is valid when itslattice constant is in the order of the mean free path ofphotoelectrons or smaller. This element assignment directlygives the element information of neighbors. On the otherhand, it is possible to guess the element assignment usingdifferent core levels. However, one must solve a complicatedpuzzle to assign the element of the neighboring atoms,because the origins of the view point are different fordifferent cores. The use of both element assignment andholograms of different elements directly leads to the correctcompound atomic arrangement.

4. Conclusions

We proposed an element analysis method for 3D atomicstructures using photoelectron holography. Both elements of

the emitter and scatterer atoms can be assigned. We appliedthis method to the measured InP hologram, and the scatteredatoms up to the fifth-nearest neighbors were assigned. Fromthe map of the atomic image shift caused by the atomicnumber difference, the scatterer element can be assignedwhen the atomic number difference is approximately morethan twofold. Note that most metallic oxides satisfy thiscondition.

Acknowledgments

This work was performed with the approval of the JapanSynchrotron Radiation Research Institute (Proposal Nos.2006B1019 and 2007A1278). The authors deeply thankDr. Fang Zhun Guo, Dr. Yukako Kato, Dr. Takayuki Muro,Dr. Tetsuya Nakamura, and Dr. Toyohiko Kinoshita for theirsupport in the experiments.

�matusita@spring8.or.jp

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