Holographic inversion of photoelectron diffraction ... - Science Direct

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Surface Science North-Holland

264 (1992) 380-390

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Holographic inversion of photoelectron from Cu( 001) A. Stuck,

D. NaumoviC,

diffraction patterns

H.A. Aebischer, T. Greber, J. Osterwalder

Institut de Physique, Perolles, Uniuersite’ de Fribourg

1700 Fribourg, Switzerland

Received

5 November

12 August

1991; accepted

for publication

and L. Schlapbach

19Yl

The electron wave field near the photoemitter is calculated from measured photoelectron angular distribution maps of CutOOl) using Huygen’s theorem. A novel formalism for these calculations is introduced. The resulting real-space images exhibit considerable structures along the nearest and next-nearest neighbour directions. By comparing images taken’at 7 different kinetic energies (768-1740 eV) it is shown that these structures cannot be readily assigned to atomic positions but are more probably related to the crystal symmetry

1. Introduction It was recently suggested by SzGke and Barton [ 1,2] that photoelectron diffraction (PD) patterns of adsorbate levels can be viewed as holograms and that real-space images of the atoms surrounding a photoemitter can be obtained by a phased Fourier transformation. This idea was applied to substrate emission by Harp et al. [3,4] who showed that the Fourier transform of experimental Auger electron diffraction (AED) patterns from Cu(OO1) and Cu(ll1) measured at a kinetic energy of 918 eV shows peaks close to the positions of the nearest neighbours and much stronger peaks between the emitter and the nearest neighbours. Several theoretical investigations involved simple structural models such as regular chains of atoms or dimers, or adsorbate systems at one energy [2,4-61. The holographic reconstruction of simulated substrate emission for different energies has been performed by Tong et al. [71. At each energy the calculation in a narrow cone around a direction of interest is used. It has been recognized that the highly anisotropic character of electron scattering at energies above 500 eV moves the peak positions, apparent in the holographic reconstruction, away from the true 0039-6028/92/$05.00

0 1992 - Elsevier

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atomic positions by as much as 30% [4-61. A combination of the strong angular dependence of the scattering amplitude as well as the angular dependence of the scattering phase is responsible for these shifts. Several procedures to compensate this effect have been proposed [4,5,71. At present the discussion of photoelectron holography gives a very optimistic prospect of a direct surface structural technique with resolutions of a few tenths of an Angstrom, a view based mainly on simulated (PD) data. However, Huang et al. and Wei et al. [7-91 have emphasized that the image reconstruction from a single-energy photoelectron hologram is unreliable in the case of substrate emission. In this paper we demonstrate experimentally that the peaks observed by Harp et al. [3,4] along the nearest neighbour directions as well as the peaks along the next-nearest neighbour directions cannot be interpreted as atoms since their positions scale with the wavelength of the electrons at energies above 768 eV. For this purpose, we have recorded a series of full solid angle PD patterns of Cu(OO1) at kinetic energies between 768 and 1740 eV. It will be shown that the PD patterns of CufOOl) in this energy range essentially depend on the crystal symmetry and forward scattering,

B.V. All rights reserved

381

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OO1)

while only the first interference maximum of the nearest neighbours can be directly identified in the data. This is not sufficient to determine the atomic positions accurately by Fourier transformation using only one kinetic energy.

2. Theory In photoelectron holography a PD pattern is viewed as an interference between that part of the photoemitted wave which propagates directly from the emitting atom into the detector (i.e., the reference wave) ?& and the part scattered from nearby atoms (i.e., the object wave) ?ZJot,,which carries the information about the atomic positions [Z]: pe, = Fr;er+ qob. We, is the total photoelectron wave function. Assuming free propagation of the electron wave one can use the Helmholtz-Kirchhoff theorem [lo] to calculate the amplitude U(r) of the object wave field at any point r in space from the amplitudes of the electron wave on a sphere S: ~2

,iklR-rl

_-ikR

A global phase factor can still be chosen arbitrarily. The emitting atom is at the origin of the coordinate system and the integration extends over all points R on the sphere of measurement S. k is the wave number of the electron and the far-field amplitude ~$0, 4) is related to the electron intensity Z(0, 4) measured in the direction 8, 4 and the intensity I, of the reference wave by

El:

xw

4) =

w-c4) -1, 6

.

(4

Eq. (1) is nothing else but Huygen’s principle [ill. Every point on the sphere of measurement is the source of a spherical wave with the far-field amplitude x(0, 4) and the coherent sum of all these waves yields the phase and amplitude of the object wave at any point in space. Since the measured electron intensity Z(fZ,$1 is proportional to 1pe, I 2, we find for ‘ZJrefreal at the site of the detector, i.e., in the far field: Z(e, 4) N

I Prer I 2 + I Yob I 2 + 21k;,, Re(V,,J. with I Pref 12 one obtains:

kientifying

IO

Pa) The measured intensity Z(e, 4) has been converted to the amplitude x(0, 4). As in optical holography, the first two terms in eq. (2a) are responsible for the formation of the reconstructed image respectively the mirror image, while the third term is a perturbation. It is neglected in optics but it will produce artefacts in the case considered here because the object and reference waves are intrinsically linked [6,12]. The image function U(r) is then just the amplitude !&, + Fz + ly,#$/!Z’~~, evaluated near the emitter, i.e., in the near field, I rU(r, 8, +)I 2 is then approximately proportional to the probability of finding a scattered electron (i.e., the object wave) at the distance r from the emitter within an infinitesimal cone dR pointing in the direction 8, 4. Consequently, if the electron scattering is not too anisotropic one can expect to find peaks near atomic positions. The theory outlined above is a scalar wave theory and assumes a free propagation of the reference wave. Therefore, the same mathematics can be used as in classical electrodynamics, always keeping in mind that in general the physics of electron waves and light waves is different. As in classical electrodynamics, the propagator in the integral of eq. (1) can be expanded into spherical harmonics Y,,(e, 4), spherical Bessel functions j,(h), and Hankel functions h,(kR), known as the multipole expansion [13]. Since only atomic distances r are of interest, the conditions r ‘il(kr) I=0

f: In=

Yrm(a, P)arm, -1 (3)

Here (r, a, fl) are the polar coordinates of vector r and CR, 0, t#~)are the polar coordinates of vec-

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OO1)

382

tor R. The integration in eq. (4) extends over a sphere of unity 0, making the formalism independent of the distance of measurement R. The same approximation, r -=KR, without expansion into spherical harmonics, leads to the spherical Fourier transformation derived by Barton [2]: U(r) = &/x(H,

4) e-ik’r do.

(5)

Both treatments are mathematically equivalent. Eq. (5) is usually numerically evaluated in a plane onto which the sphere of unity fi is projected [21. This step is avoided by our method which solves the problem on the sphere. As will be shown below the multipole expansion further allows us to quantify analytically the relationship between the angular step size of the measurement and the distance from the emitter up to which one can expect to see atoms. Because of j, = sin x/x, the term with I = 0 in eqs. (3) and (4) describes the diffraction-limited image of a point source at the site of the photoemitter whose weight a,, is just the average of ~(0, 4). Generally the position of the first and strongest maximum of j,(S) shifts to higher arguments a,,, as I increases. For 5 < I< 100, the relation is approximately linear: 6,,,(l)

= 2 + 1.

(6)

Coefficients a,, with small 1 thus contribute predominantly to the image near the emitter while the coefficients with higher I form the image further away. Consequently, if we restrict the image to a volume smaller than a sphere of radius the sum over 1 in eq. (3) can be truncated at z?ka_ where krmax= 6,.&,,). Since experimental XPD patterns are only measured in a number of discrete directions, the finite sampling density in k-space will limit the volume around the emitter in real space which in principle can be seen by applying the transformation. The fastest oscillations of Y,,(B, 4) are proportional to eime, respectively, eim9, while according to the sampling theorem the fastest significant oscillations in a discrete Fourier transform connect two adjacent points with a half wave [14]. If

A0 = A4 is the typical angular step size of the measurements one gets the condition: A81,,, = rr.

(7)

Combining eq. (6) with eq. (7) then yields a distance rmax in real space up to which one can expect to see atoms. For the CuLVV Auger at 918 eV and AB = 5” one gets a value of rmax= 2.5 A, which is close to the nearest-neighbour distance in Cu. An improvement of the angulzr resolution to A8 = 2” increases rmax to above 6 A. These estimates are in excellent agreement with simulations along atomic chains by Harp et al. [15]. In the experiments shown below we have Ae = 2” and thus I,,, = 4.3 A for E = 1740 eV, the highest kinetic energy used in this study. For all energies shown, the inelastic Dmean free path of the electrons is larger than 8 A, and thus ate can expect to see at least the nearest (; = 2.55 A) and next-nearest neighbours (r = 3.61 A) of Cu in the real-space images generated from the data. If only on the radial distribution U(r) along a given direction (Y,p is of interest one can choose a new coordinate system in which this direction points along the z-axis. We use primed symbols to denote angles in this new coordinate system. The given direction (Y, p is then represented in the new coordinate system by LY’= /?’ = 0. Since Y,,(O, /3’> N &,m/w, the sum over m in eq. (4) vanishes and eqs. (3) and (4) can be simplified: U(r) =2C(21+1)(-i)‘j,(kr)b/,

sin 8’ de’P,(cos

b, = $

e’)f(e’),

0

f(er)

=

&izTd+'x(e',

4’).

(9) (10)

Thus, as already discussed by Saldin et al. [161, the radial distribution of U(r) in a given direction (Y, p depends only on the average of xc@‘, 4’) over all azimuthal angles $‘, the one-dimensional function f(e’). By performing this integration around a given direction the diffractive, and the nondiffractive or forward-focusing origin of the features in the data can be discussed before carrying out the transformation (eq. (8)). If the z-axis

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OO1)

is chosen along an inter-atomic direction, diffractive features, caused by interference, should shift to higher polar angles 8’ as the energy decreases [17], whereas nondiffractive features are linked to the crystal symmetry and should not move considerably. For kr x=- I the Zth spherical Bessel function can be approximated by [131 j,(

kr)

=

sin(

kr -

lr/2)/kr.

1

cos

et)

x(0 + r, 4), i.e., point symmetry. To save computer time we have further assumed x(0, 4) = x(0, 4 + r>, i.e., twofold symmetry of the crystal with respect to the surface normal. This condition is clearly fulfilled for Cu(OO1). Only the coefficients al,, respectively, b,, with even 1 and m have then to be taken into account.

3. Experimental

As will be shown below, only even I’s were used in this study to calculate the images. Consequently, one part of the contribution of the fth multipole component to the image function proportional to I rm, 8, 4) I 2 is then I b, I 2 sin2(kr). Far away from the emitting atom one will thus simply have the square amplitude of an outgoing wave. Near the emitter the spherical Bessel functions with small l’s will already contribute fast sin2(kr) oscillations to the image function. To suppress these contributions we have convoluted the function &U(r, 8, I$) with a Gaussian (FWHM = 0.U) before and after squaring. It was verified that this procedure did not change the peak positions of the envelope but smoothed the fast sin2(kr) oscillations. In this way all radial plots presented here were treated. The image shown in fig. 2 however, was not smoothed. Once the multipole coefficients a,, representing the data set x(f3, 4) have been numerically calculated, the function f(0’) can be expressed as follows:

fw =4KDl(

383

f:

ylm(~, i+h

m=-1

(11) We have numerically evaluated f with conventional integration algorithms using eq. (10) as well as with an algorithm based on eq. (11). Both schemes yielded essentially the same results which places confidence in the formalism and the numerical implementation. Since PD patterns can only be measured above the crystal, the far-field amplitude ~(0, 4) has to be extended to the lower hemisphere. In the present work we have assumed x(0, 4) =

The experiments were performed with a modified VG ESCALAB Mark II spectrometer, equipped with a three-detector unit which enabled the data accumulation at a high angular resolution down to a full acceptance cone of A0 < 1”. The angle scanning is fully automated and done in an azimuthal fashion: The polar angle is first fixed at 8,, = 78” and is reduced by A8 = 2” after each full azimuthal rotation. The azimuthal step size is increased to A4 = towards lower polar angles to give an (se/e)a,, almost uniform sampling density across the hemisphere. In this way about 3500 angular settings are scanned with an accuracy of about 0.2”. At each setting, the signal is extracted from a photoelectron spectrum by a curve fitting procedure [181. The signal as well as the corresponding background are then recorded. The orientation of the sample was better than 1” and its cleaning was done in situ by repeated cycles of Ar+ sputtering up to 1.5 keV and subsequent annealing up to 1200 K. The crystalline order of the surface was checked by LEED. Scan times varied from 12 to 60 h during which the surface always collected less than 20% of a monolayer of oxygen and carbon at a typical pressure of 5 X lo-” mbar. All points shown in the two-dimensional intensity maps (fig. 1) are measured and no symmetry operation of any kind was performed on the data. In order to minimize the effects which are due to instrumental response we normalize the background signal B(B, 4) and the photoelectron intensity Z(e, 4). The experimental x(0, 4) is then defined as the difference between these two normalized distributions: / dRZ(8,$)

= / d&!B(B, f$) = 1,

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OOli

384

and

The definition a oo =

/

dfix(o,

above means 4)

= 0,

that (12)

which eliminates in the image all contributions caused by the point-emitter at the origin. Combining eq. (12) with eq. (2) yields / dRZ(0, 4) = 47~1,, i.e., the total measured electron flux is equal to the total electron flux of the emitter. This implies that the reference and object waves are equally damped within the solid. At the high kinetic energies used, this assumption is not unreasonable. However, at lower energies, when the inelastic mean free path is comparable to the nearest neighbour distance, eq. (12) can be wrong. In order to be able to measure photoelectron intensity maps at as many energies as possible, the sample was excited with MgKa and SiKa radiation. We then recorded different photoelectron peaks as well as Auger lines. A summary is given in table 1.

4. Results

and discussion

Three selected PD patterns are shown in fig. la-lc in the stereographic projection. The intensity distributions are dominated by the forwardfocusing peaks along (011) and (001) and the Kikuchi bands which connect these forwardfocusing directions. The latter are centered around the projections of high-density crystal planes and are also apparent in the diffraction patterns of AltOOl) and Pt(ll0) [19]. The modulations along these bands are more pronounced at lower energies while the forward-focusing peaks and the bands broaden. The eight-leaved flower patterns around (001) do not shift as a function of energy while the Y-shaped intensity enhancements along (111) turn into doughnut-like rings at kinetic energies lower than about 1000 eV [20]. Both the flower patterns around (001) and the Y-shaped intensity enhancements along (111) are also observed for AltOOl) and Pt(ll0) [19]. Fig. Id

shows the stereographic projection of dense crystal planes and some forward-focusing directions. As a check of our Fourier transformation method, a calculated real-space cut of I C/(r) I * perpendicular to [ 1111 through the nearest neighbours is shown in fig. 2 (left-hand side) Hardcastle et al. [4] published an image along this cut obtained from a Cut1111 crystal. In agreement with them we observe maxima near the atomic positions. However, other structures appear in their as well as in our images. Since our measurements were taken from a Cu(OO1) crystal, the instrumental response function, mean free path effects and the cut-off of the data at grazing angles break the symmetry of the image in fig. 2. An additional check of our multipole expansion formalism is shown in fig. 3. Radial image functions obtained by inverting the calculated diffraction patterns of two Cu dimers at 918 eV, using a single-scattering-cluster code described elsewhere [21], are compared with the transformed image of the structure factor eikdcosO. The image from the structure factor is a single peak located at the nearest, respectively, next-nearest neighbour distance and thus confirms again the correctness of the calculation. The forwardscattering and phase shift anisotropies caused by the strong atomic potentials seriously distort the images of the more realistic dimer calculations. The fact that away from the scattering atom constant means that I i-U(r) I 2 is approximately electrons are focused along the axis of the dimer. At kinetic energies around 1 keV the peaks can shift from their correct positions by as much as 30% [4-61. Similar effects are expected in transformed images of the raw data measured at high kinetic energies. For geometrical reasons the atoms further away from the emitter are less illuminated by the reference wave than atoms near the emitter. The factor r in I rU(r>12 compensates this. In such a plot the peak heights are a measure of the scattering probability of the atoms, as long as the inelastic attenuation of the reference wave remains small. The &integrated functions f(f3’) of eq. (101 along the [OOl] direction are shown for seven different energies in fig. 4. All the curves are very similar. One can well distinguish the forward-

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OO1)

385

Fig. 1. XPD maps of different Cu emissions. The pictures are stereographic projections of the measured intensities. The big circle corresponds to grazing emission while normal emission is in the center of the image. Various forward scattering peaks and Kikuchi bands are visible. (a) Cu3p, E = 1664 eV, (b) Cu Auger LW, E = 918 eV; (cl Cu Auger LW, E = 768 eV. (d) Stereographic projection of dense crystal planes. The black dots indicate the location of some forward scattering directions. The (001) circle corresponds to grazing emission.

A. Stuck et al. / Holographic inversion of PDpattems from Cu(OO1)

386

Fig. 2. (Left-hand side) Real-space image I U(r) I * normal to the [ill] direction in a plane containing the nearest neighbours. The expected atomic positions are marked by crosses. The symmetry is broken because the crystal is oriented normal to [OOl]. A drawing of the cut is shown on the right-hand side. The kinetic energy of the electrons was 918 eV. (Right-hand side) Drawing of the (111) plane shown in fig. 2 in relation to the crystal lattice. The emitter is behind the plane and the surface normal is [OOl].

focusing maximum along [OOl] and several maxima near 21”, 38” and 53” off normal. The first maximum between 20” and 2.5” corresponds to the eight-leaved flower pattern, while the second maximum between 35” and 40” is mainly due to the Kikuchi line along the (Ill)-planes. Since the forward-focusing peaks along the (011) nearest neighbour directions are rather narrow their contribution to the integral is small and at 45” no

[OOl]dir.

I

I:Oll]dir.

Table 1 Summary of all electronic transitions used in this study with the kinetic energies measured and the excitation used Kinetic (eV) 1740 1664 1250 1178 918 807 768

energy

Transition

Excitation

I

0.1

r 0.0

& 1.0

2:o

3.0

4.b

5.0

6.

r [AI

Valence band cu 3p Valence band cu 3p LVV Auger cu 2p LVV Auger

Si Si Mg Mg Mg Si Mg

Fig. 3. Transformation of a Cu dimer oriented in the [OOI] direction with the scattering atom at the next-nearest neighbour distance (d = 3.61 A, top) and a cluster containing 4 equivalent scatterers at the nearest neighbour positions (d = 2.55 A, bottom). The small lines indicate the correct atomic position. Note that the radial probability distribution I rU(r)l’ is plotted. (a) Transformation of the structure factor e’kdc”‘H alone. (b) Single-scattering calculation for Cu at E = 918 eV.

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OO1)

maximum is observed. Finally, the Y, respectively, doughnut pattern around (111) causes the modulations between 50” and 55”. However, all these structures do not shift as a function of energy and the holographic information cannot easily be identified. This is reflected in the transformed functions U(r) shown in fig. 5, calculated along [OOl]. While at each energy many peaks are visible in these images their position does not change when they are plotted against the distance r/h, i.e., the distance measured in units of the electron wavelength A (fig. 5a). Consequently, they are due to the crystal symmetry and are not caused by interference. Fig. 5b emphasizes this point; only at two similar energies, E = 1664 eV, respectively, E = 1740 eV, do we find maxima near the correct location of the next-nearest neighbours. For the nearest neighbours the situation is slightly different, as is shown in fig. 6. Here, the

c

cu (001): [OOI] direction

1740eV

4ev

125oev 17aev

\

\

I

10

’

I

20

‘,I

V 30 8 '

I 40

‘I’,’ 50

60

IC 70

[degrees]

Fig. 4. The function f(S’) evaluated around [OOl]for different kinetic energies. The small lines indicate at every energy the expected positions of the first interference minimum, respectively, maximum according to single-scattering theory.

r 14 Fig. 5. (a) The image I rU(r)l ’ along [OOll in real space for different kinetic energies. The distance is measured in units of the electron wavelength A. (b) The image IrU(r)12 along [OOl] in real space for different kinetic energies, the distance is measured in angstrom. The correct location of the nextnearest neighbours at 3.61 A is indicated by the dashed line.

$-integrated curves f(e’> along the [Oil] direction are plotted. As in the previous case most structures do not depend on the energy. However, interference structures close to the (011) directions can be distinguished. The minimum next to the forward-focusing peak along [Oil] shifts from 13” at the highest energy to 18” at 768 eV, and the position of the first maximum changes from 25” to 30” as the energy decreases. Contrary to the expected interference effects the other two prominent structures, around 45” and between 55” and 60” move slightly towards the [Oil] direction as the energy decreases. The forward-scattering peak and the two Kikuchi bands through the normal form the structure around 45” while the last modulation is probably caused by the eightleaved flower pattern. The results of the same analysis as before are presented in fig. 7 for the [Oil] direction. At energies below 1 keV two double-peak structures can be well distinguished: One at r/h - 3 and one at r/A - 7. The peaks of the second structure stay precisely at the same

A. Stuck et al. / Holographic inwrsion of PD patterns from Cu(OO11

388

positions when measured in units of the wavelength (fig. 7a). Above 1 keV the peak at r/A 7.5 remains while a strong peak emerges at r/h - 5.5. Around 1.2 keV a strong modulation is seen at r/A - 9.5. Plotted against the absolute distance only one of those peaks, at 918 eV, is found precisely at the nearest neighbour distance of 2.55 A. At 807 and 768 eV one finds two peaks shifted slightly outwards from the correct position. However, above 1 keV no strong peaks are found near the atomic positions. The nature of the peaks between the emitter and the nearest neighbours is clearly not diffractive (see fig. 7a) and was attributed to forward scattering by others [41.

One might argue that the peaks seen between 2.5 and 3 A at energies below 1 keV do indeed correspond to atomic positions. As already discussed above, deviations from the correct positions are expected and caused by anisotropies of

10

5

rlh Fig. 7. (a) The image / d/(r)/ ’ along [Oll] in real space t’o~ different kinetic energies. The distance is measured in units of the electron wavelength A. (b) The image 1d/(r)1 ’ along [Ol I] in real space for different kinetic energies. the distance is measured in ingstriim. The correct location of the nearest neighbours at 2.55 A is indicated by the dashed line.

cu (001):

[Ol l] direction

h

1740ev

16640V

125Oev

117aev 91sev 8070V

s

P-----I,

I,

10

I

20

,

30 8 '

I

,

40

I,

I

I,

50

60

,

I

70

[degrees]

Fig. 6. The function f(0’) evaluated around [Oil] for different kinetic energies. The small lines indicate at every energy the expected positions of the first interference minimum respectively maximum according to single-scattering theory.

the electron scattering. We would like to emphasize however that the structures in fig. 7 arc found at constant position when plotted against r/A, indicating a symmetry effect. Furthermore, while the data at 768 and 918 eV correspond to Auger transitions, the diffraction at 807 eV was measured with Cu2p photoelectrons. Hence the different initial states associated with these three measurements are not important. At the high energies discussed here the first interference maximum is generally about half an order of magnitude smaller than the forwardfocusing intensity. The Fourier transformation gives the best results if oscillations extend over the whole sphere. Consequently, the first interfcrence minimum and maximum seen along [Olll are not sufficient to determine the bond length with Fourier techniques, i.e., if the interference fringes around a given direction are contained within a cone of about 30”, the resolu$on of the Fourier transformed image is about 3 A at 1 kcV kinetic energy [7].

A. Stuck et al. / Holographic inversion of PD patterns from Cu(OO1)

At angles larger than about 30”, the higherorder interference fringes are strongly dominated by the eight-leaved flower pattern, the Kikuchi bands and other forward-focusing peaks. The eight-leaved flower pattern around [OOl] is at least partly formed by the forward-focusing peaks along the (012), (113) and (114) directions while the Kikuchi bands can be described as Bragg reflections of the electrons along low-index planes. All these effects mainly depend on the crystal symmetry whereas their sensitivity to distances is small. The transformation is expected to give significant results if these effects are precisely eliminated. At the same time the positional errors due to the scattering anisotropies will be suppressed. For this purpose different procedures have already been proposed, such as modifying ~(0, 4) functions or integrating over multiple energies [4,5,7]. At present it is not clear which procedures work best. It has also been shown theoretically that self-imaging of atoms is present in the transformed images [6,12]. This tends to reduce the resolution and can completely destroy any information about atomic positions in the case of nonperiodic structures 1121. At kinetic energies below a few hundred eV the forward-focusing peaks and the Kikuchi bands vanish but the resolution is reduced since the electron wavelength is of the order of one Bngstriim or more. Additionally at these energies initial-state effects can be important and have to be included in the algorithms.

5. Conclusions

We have developed a novel Fourier transformation technique to calculate real-space images out of PD patterns. The accuracy of the inversion of the dimer models and the fact that f(0’) of eq. (10) does not depend on the algorithm used, give confidence in the numerical implementation. Although reconstructed images taken at 918 eV show peaks at the correct positions they scale with the electron wavelength and are probably caused by the crystal symmetry. Little indication of nearest neighbour distances was found in the

389

transformed images. Most problems are caused by forward-focusing and Bragg scattering. One can still hope to extract correct atomic positions by holographic techniques if these effects are eliminated theoretically or experimentally, for example by taking into account multiple energies or by choosing adsorbate systems.

Acknowledgements

We would like to mention the skilfull technical assistance of F. Bourqui, 0. Raetzo and H. Tschopp. We are also grateful to N. Montini for correcting our English. This work was supported by the Swiss National Science Foundation.

References

111A.

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390

A. Stuck et al. / ~o~ogru~~ic aversion of PD patterns from Cu(OOf)

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