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Direct surface-structure analysis by the peak rotation in circularly polarized light photoelectron diffraction
Surface Science 471 (2001) 143±150
www.elsevier.nl/locate/susc
Direct surface-structure analysis by the peak rotation in circularly polarized light photoelectron diraction Hiroshi Daimon a,*, Shin Imada b, Shigemasa Suga b a
Graduate School of Materials Science, Nara Institute of Science and Technology 8916-5 Takayama, Ikoma, Nara 630 0101, Japan Department of Material Physics, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560 8531, Japan
b
Received 4 July 2000; accepted for publication 1 October 2000
Abstract Rotation of the forward-focusing peak positions in the circular dichroism in photoelectron diraction (CDPD) pattern is discussed in detail. Based on this phenomenon a new method of direct analysis of three-dimensional surface structure is proposed. The direction of the forward focusing peaks in CDPD contains information not only of the direction but also of the distance between the source atom and the scatterer atoms. Thus the three-dimensional surface structure can be directly determined without any complicated calculation. A formula for the angular dependence of eective angular momentum, m h, of photoelectron has been developed for the analysis. The typical accuracy in this which is comparable to usual photoelectron diraction analysis and is enough to make a analysis is around 0.25 A, starting model for more detailed analyses such as LEED. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Photoelectron diraction measurement; Photoelectron diraction; Tungsten; Low index single crystal surfaces
1. Introduction Determination of surface structures is still a dicult problem because of the lack of periodicity in the surface normal direction. So far have been developed many methods, such as surface X-ray diraction, X-ray standing wave method, impact collision ion scattering spectroscopy, low energy electron diraction (LEED), re¯ection high energy electron diraction (RHEED) and so on for this purpose. Among these methods the photoelectron holography, [1±3], which is a recent extension of the photoelectron diraction, has attracted much * Corresponding author: Tel.: +81-743-72-6020; fax: +81743-72-6029. E-mail address: [email protected] (H. Daimon).
attention because it can in principle reconstruct the three-dimensional structure around a speci®c atom without the so called phase problem. It has been recognized, however, that the inversion method (Fourier transformation) to obtain the real structure is not so straightforward. The diculty in the photoelectron holography comes mainly from the strong forward focusing peaks [4] in two-dimensional photoelectron diraction patterns, which always appear when the emitter atom is below the scatterer atoms. Many eorts have been made to eliminate the disturbance from these forward focusing peaks by means of SWIFT [5,6] or SWEEP [7±9]. In general, a forward focusing peak has information about the direction from the emitter to the scatterer but does not have the information about
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H. Daimon et al. / Surface Science 471 (2001) 143±150
the distance between them. Recently, a strong circular dichroism in photoelectron diraction (CDPD) was found for non-chiral and nonmagnetic materials [10,11]. The forward-focusing peak-positions in the two-dimensional photoelectron diraction pattern excited by the circularly polarized light rotated in the same direction as the rotation of the electric vector of the light. This rotation angle being calculated by a simple formula including the angular momentum, m, of the photoelectron could reproduce the experimental peak positions with reasonable accuracy. Using modi®ed version of this formula we can directly construct three-dimensional atomic structure by measuring the azimuthal shift of the forward focusing peaks in the photoelectron diraction patterns excited by the circularly polarized light. Hence, this method positively utilizes the forward focusing peaks for the structural analysis. The eective m value, which is an average of possible ®nal-state m values at certain polar angle, is discussed in detail for the analysis. The applicability of this analysis is discussed in a typical example of W(1 1 0)1 1-O surface [11]. 2. Example Fig. 1 shows an example of two-dimensional diraction pattern from the chemically shifted W 4f peak from the ®rst-layer W atoms of W(1 1 0)1 1-O surface [10]. The stereographic
Fig. 1. Two-dimensional diraction patterns of the chemicallyshifted W 4f oxide peak from the ®rst layer W atoms. (a) and (b) shows the results for LP and LCP, (ccw) light, respectively. The white lines show the directions of the forward focusing peaks observed in the LP pattern.
projection method was used to plot these ®gures. The center of each pattern corresponds to the surface normal direction and the horizontal and vertical edge corresponds to the polar angle of 90°. The photon was incident normal to the surface. The polar take-o angle range is from 61.0° to 73.5°, which is almost the same as that over which the forward focusing peaks in XPS [12] are observed. The photoelectron kinetic energy is 317 eV. Fig. 1(a), and (b) show the results for linearly polarized (LP) and left circularly polarized (LCP) light, respectively. Here the de®nition of LCP is that the rotation direction of the electric vector of the photon in the planes of Fig. 1(b) is counterclockwise (ccw). We can see basically six peaks in each of the pattern of Fig. 1. These strong peaks in the photoelectron angular distribution pattern at the kinetic energy above 200 eV is recognized as forward focusing peaks [4]. When the excitation light is not circularly polarized (namely, LP or unpolarized), the forward focusing peaks appear along the directions connecting the scatterer-atoms and the photoelectron-emitting atom. In this case the emitter atom is the top layer W atom and the scatterer atoms are O atoms above it. Hence the forward peaks in Fig. 1 are considered to be dominated by single scattering. The azimuthal angles of the peaks for LP as measured relative to the [0 0 1] direction are 28°, 90°, and 152°, which are indicated by white ®ducial lines. These angles correspond reasonably closely with the angles of the forward scattering peaks observed in high-energy XPD [12]. It is clear that there is a tendency for the pattern to shift counterclockwise with LCP excitation in Fig. 1(b). Such peak ``rotations'' are of the same qualitative type as seen by Daimon et al. for Si(0 0 1) [10] for bulk photoelectrons. The circles and squares in Fig. 2 are the phi dependence of the averaged peak intensities for LCP and LP, respectively. The LP data has been averaged using mirror symmetry with respect to 1 1 0 direction, which is 90° here. The vertical solid lines show the average positions of the peaks in the LP pattern in Fig. 1(a), and we denote these peaks by A, C, and B. The rotation angle of the peak A and C in LCP data is 12° and 14° as shown by the broken lines in Fig. 2.
H. Daimon et al. / Surface Science 471 (2001) 143±150
145
Fig. 2. The azimuthal dependence of the W 4f oxide peak intensity in Fig. 1 as averaged over polar angles. The LP data have been averaged using mirror symmetry with respect to 1 1 0 direction, which is located at 90° here. The vertical solid lines show the positions of the forward focusing peaks observed in the LP data. The vertical broken and dotted lines show the estimated positions of the peaks in the LCP data calculated by Eq. (1) using m m and 4, respectively.
The rotation angle D of the peak around the photon incident axis is reproduced well by the simple formula [10] m m : 1 D tanÿ1 2 kR sin h kR sin2 h Detailed theoretical works [13±15] support this formula. Here, m is the magnetic quantum number (z-component angular momentum) of the photoelectron, k is its wave number, R is the internuclear distance between the emitter and the scatterer, and h is the angle between the photon incident direction and the outgoing photoelectron direction. When the position vector of the scatterer is described as R; h; / the peak position observed by using cw and ccw circularly polarized light appears at h; / D. This relation is shown in Fig. 3(a).
3. Eective m value First we consider the transition from an initial state Wi with the quantum number of the orbital angular momentum l0 and its z-component m0 as Wi Rnl0 rYl0 m0 h; /;
2
where Rnl0 r is a radial wavefunction and Yl0 m0 h; / is a spherical harmonic. In the transition caused by a photon, the spin is neglected because it
Fig. 3. Explanation of the rotation angle D using the constant phase surface.
does not change. The quantum axis (z-axis) is the travelling direction of the incident circularly polarized light. The ®nal state wave function Wf of the emitted photoelectron at r outside the emitting atom, which is the original wave in photoelectron diffraction, is expressed as Wf /
l X X
alm Rkl rYlm h; /;
3
ll0 1 mÿl
where Rkl r is an outgoing radial spherical wavefunction, k is the wave number of the photoelectron, and alm is a constant to be determined by a transition matrix element shown below. The transition operator r e in the circularly polarized-light photoexcitation process within the dipole approximation is conveniently expressed by means of the spherical harmonics. The transition operator of any polarization can be expressed as r 1 X 4p el Y1l h; /r; 4 re 3 lÿ1
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H. Daimon et al. / Surface Science 471 (2001) 143±150
ÿ p where e1 ex iey = 2 and e0 ez . For a circularly polarized light propagating along the +z direction with spin k 1ÿ1, the electric vector of the photon e eÿ is expressed as eÿ1 ; e0 ; e1 0; 0; 1 ÿ1; 0; 0. The transition matrix element is written as [16] Z r e Wi d3 r alm Rkl Ylm X ÿ 5 Rnl0 !kl c1 lm; l0 m0 ; ll0 1
where Rnl0 !kl is a radial matrix element, and c1 lm; l0 m0 is the Gaunt coecient. 1 This c1 lm; l0 m0 is not zero only when m m0 1. Hence, a circularly polarized light with the angular momentum of photon k of 1 excites the ground state with the magnetic quantum number m0 to the ®nal state with m m0 1. In other words, the angular momentum that the electron gains by photoexcitation is 1. The ®nal state wavefunction Wmfk excited by a photon of k from l0 ; m0 core is written as X ÿ Rkl rYlm h; /Rnl0 !kl c1 l; m; l0 ; m0 Wmfk / ÿ
ll0 1 0
mm k :
6
7
First we consider a ®nal state with only one m component. Since we are considering the photoelectron, we can omit the incoming wave in Rkl and assume that the ®nal state wavefunction outside the emitting atom is expressed as W/
eikr Ylm h; /; r
8
W/
eikr Hlm heim/ ; r
9
where Hlm h is the h-function in the spherical harmonics.
1
ck lm; l0 m0 sinh dh d/.
q R R 4p 2k1
expi kr m/:
10
Consider the phase at A, B, and D in Fig. 3(b) within a small region around the scatterer A. The plane of Fig. 3(b) includes O, A, and y-axis. Hence r direction is h degrees inclined from z axis. The quantities in are those projected on the plane perpendicular to the z axis, and not in the plane of Fig. 3(b). The phase dierence between A and B is described by the azimuthal dierence between A and B multiplied by m, i.e. m AB=R sin h. The phase dierence between D and B is kBD. The phase is constant anywhere on the line AD when m AB=R sin h kBD. This line is the wave front, and the propagating direction of the wave is perpendicular to it. Hence the shift of the forward direction D0 from OA direction is expressed as tan D0 BD=AB
m : kR sin h
11
D0 is converted to D from Fig. 3(a) as
In a special case of s core excitation, Wfk / Rk1 rY1k h; /:
The propagating direction of this ®nal state wavefunction is intuitively calculated considering the wave front of Eq. (9), which is a simple and dierent derivation from Ref. [10]. The phase of the wave is expressed as
Yl m h; /Yk;mÿm0 h; /Yl0 m0 h; /
R tan D0 OE R sin h tan D:
12
From Eqs. (11) and (12), we obtain Eq. (1) as D tanÿ1
m m : 2 kR sin h kR sin2 h
13
The physical base of the azimuthal shift is summarized as follows. The forward focusing peak is the ``zeroth order diraction'' peak, which appears because the phases of all wavelets on the wave front being scattered by each in®nitesimal atomic potential coincide in the forward direction. This ``forward direction'' is perpendicular to the wave front of the wave. When the wave has an angular momentum, the wave front hits the scatterer at an angle. Hence the forward direction of the photoelectron wavefunction is inclined at the scatterer. In other words, the peak does not shift from the forward direction but the forward direction has shifted already before the scattering.
H. Daimon et al. / Surface Science 471 (2001) 143±150
This formula for the direction of the forward focusing peak is exact when the photoelectron wavefunction can be expressed by only one component of angular momentum. This condition is satis®ed in the excitation of s core, where ®nal m should be 1 as shown in Eq. (7). In the following, we consider more general case of excitation. Usually, the l0 1 component prevails in the ®nal state compared with the l0 ÿ 1 component. For example, we consider the case of the photoexcitation from the Si 2p l0 1 core state by a circularly polarized light of k 1. This case is for the result of Ref. [10]. In this case, the ®nal l is either 2 or 0, and the ®nal l; m 2; 2, 2; 1, 2; 0 and 0; 0. The probability of realizing these three l 2 ®nal states is much higher than that for l 0, and is about 13:1 at the photon energy of around 350 eV [16]. The probabilities of realizing these three ®nal m values are not equal because the Gaunt coecients are dierent. For example, the probabilities for l; m 2; 2: 2; 1: 2; 0 are 6:3:1. When k is ÿ1, the ratio for m ÿ2; ÿ1; 0 is also (6:3:1). Even when the spin±orbit interaction of the core state is considered, this ratio is unchanged as (6:3:1) for ``each'' spin±orbit split component P1=2 and P3=2 . This ratio, however, is that of the total cross section over 4p steradian and it changes when the emission angle h changes (but does not change even when / changes). In the following we develop a formula for eective value of m, m h as a function of h, considering the spin±orbit interaction. 3.1. h dependence of the expectation value of m For simplicity, we show only the case of P1=2 component, but the result is the same in the case of P3=2 component, and hence is the same for the sum of them (total 2p emission). We use the notation of j j; jz i for the state having the quantum number of
m h
12jH2;2 j2 3jH2;1 j2
the total angular momentum j and its z-component jz . For k is 1 the ®nal state photoelectron wave function from P1=2;1=2 is j3=2; 3=2i, whereas that from P1=2;ÿ1=2 is the combination of j3=2; 1=2i and j1=2; 1=2i. The transition matrix element from the core state j1=2; 1=2i to the ®nal states j3=2; 3=2i, and from the j1=2; ÿ1=2i core state to the j3=2; p1=2 i states are calculated as p1=2i and j1=2; 1= 3, 1=3, and ÿ 2=3, respectively, considering the Gaunt coecient. Then the ®nal state photoelectron wavefunctions Wf excited from the P1=2;1=2 and P1=2;ÿ1=2 states by k 1 light are expressed as ! r 1 1 4 Y2;2 b ; Wf P1=2;1=2 / Rk2 rc2 p p Y2;1 a ÿ 5 5 3 14 ÿ
Wf P1=2;ÿ1=2
2
! r r 2 3 Y2;0 a ÿ Y2;1 b 5 5 p 2 15 Y0;0 a; Rk0 rc0 3
1 / Rk2 rc2 3
where cl is the radial integral from the p initial state to the l ®nal state. a and b are up and down spin functions. We de®ne the ratio of c0 to c2 as C, and the dierence between the phase of the ®nal state (d0 and d2 ) as U. In the energy range of Ref. [10] hm 350 eV, these values C and U are estimated by interpolation of the table of Goldberg et al. [16] as 0.277 and 2.937 rad, respectively. Because the contribution from l 0 ®nal state is weak, we assume that Rk2 r Rk0 r for simplicity, which causes a small change of result through the change of U. Then the expected value m h of the angular momentum at h, is calculated by operating the angular momentum operator lz to Eqs. (14) and (15), and by normalizing it by the total excitation cross section as
p ; 6jH2;2 j 3jH2;1 j jH2;0 j2 5C2 jH0;0 j2 ÿ 2 5CH2;0 H0;0 cos U 2
147
16
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H. Daimon et al. / Surface Science 471 (2001) 143±150
18 sin4 h 18 sin2 h cos2 h
m h
2
9 sin4 h 18 sin2 h cos2 h 3 cos2 h ÿ 1 4C2 ÿ 4C 3 cos2 h ÿ 1 cosU
These formulae of Eq. (16) or Eq. (17) are the same as those obtained assuming three initial states of the form of Eq. (2), e.g. Y1;ÿ1 ; Y1;0 and Y1;1 . The m h for general l initial state can be calculated easily in a similar way as l0 P 0
0
:
17
4. New method of direct structure analysis Here, we propose a new method of direct determination of three-dimensional surface structure using above formula. The structure analysis can be
2 m c1 l0 1; m; l0 ; m0 Hl0 1;m cl0 1 eidl0 1 c1 l0 ÿ 1; m; l0 ; m0 Hl0 ÿ1;m cl0 ÿ1 eidl0 ÿ1
m h m ÿl 2 ; l0 P 1 0 id id 0 0 0 1 0 0 0 0 c l 1; m; l ; m Hl0 1;m cl0 1 e l 1 c l ÿ 1; m; l ; m Hl0 ÿ1;m cl0 ÿ1 e l ÿ1
18
m0 ÿl0
where m m0 1 for rhm 1. Usually the cl0 1 is much greater than cl0 ÿ1 . Then Eq. (18) can be simpli®ed as l0 P 0
0
2 m c1 l0 1; m; l0 ; m0 Hl0 1;m
: m h m ÿl l0 P c1 l0 1; m; l0 ; m0 Hl0 1;m 2
19
m0 ÿl0
The angular dependence of the contribution of each m state in the ®nal state, which is 2 jc1 l0 1; m; l0 ; m0 Hl0 1;m j , is shown in Fig. 4(a) and (b) for l0 1 and 3, respectively. The eective m value m h of Eq. (19) is also shown in Fig. 4(a) and (b). More strict eective m value of Eq. (18) is also shown in Fig. 4(a) denoted by ms for the case of Si 2p photoelectron at hm 350 eV. The neglect of l ÿ 1 channel causes an error of a few percent. For the case of this Si 2p example, at polar angle around h 90°, ms of Eq. (18) is 1.95, which is close to the value 1.8 of m of Eq. (19) neglecting the s ®nal state. Although the Eq. (18) is more strict than Eq. (19), it depends on both the kinetic energy of photoelectron and initial atomic orbital. Eq. (19) is convenient because it depends only on the initial angular momentum l0 and does not depend on energy or atomic orbital.
done as follows. If the forward focusing peaks in the CDPD patterns are observed at h; / D directions for ccw and cw helicity light, one knows the direction of the scatterer h; / as the midpoint of these two peaks. Then R can be calculated using m h of Eq. (19) (or Eq. (18)) as R
m h : kD sin2 h
20
Hence, the three-dimensional position of the scatterer R; h; / can be directly determined. Here we apply this method to the case of Fig. 1. The photoelectrons are from the W 4f core level, and the outgoing g or l 4 channel is expected to be dominant due to its larger radial matrix element. The angular dependence of the contribution 2 of each m state jc1 l0 1; m; l0 ; m0 Hl0 1;m j shown in Fig. 4(b) indicates that the m 4 state makes the strongest contribution between 68° and 112°. Hence the peak shift was calculated using the value of m 4 to be 15.9° for peak A and 18.6° for peak C in Ref. [11], which are shown by dotted lines in Fig. 2. However, this ®gure shows that the shifts calculated using m m h of Eq. (19) give better agreement as shown by broken lines. The result of the estimation of internuclear distance R by Eq. (20) is summarized in Table 1.
H. Daimon et al. / Surface Science 471 (2001) 143±150
149
(dD ) is between 0:3° and 1:5°. The relationship between dR and dD is dR
Fig. 4. Polar angle dependence of each m component, i.e. the square of the polar part of spherical harmonics (Hlm ) multiplied by the Gaunt coecinent (GC) for the photoelectron from (a) initial p orbital and (b) f orbital. Eective m values as a function of h is also shown. The scale of the vertical axes shows the value of m , and each m component is shown relatively.
The accuracy of determination of the azimuthal direction of the forward focusing peaks limits the accuracy dR of the internuclear distance R. The typical accuracy in estimation of peak position
R2 k sin2 h dD : m
The error values in Table 1 is estimated for m m , dD 1:5°. This accuracy dR is comparable to the usually performed photoelectron diraction analysis. It must be emphasized that this analysis does not require trial-and-error analysis process but still can directly determine the three-dimensional structure. We recommend that the kinetic energy of photoelectron for this kind of analysis should be between 200 and 500 eV. As the kinetic energy decreases, D becomes large and the accuracy increases, but the forward focusing peak blurs, which makes this analysis dicult. High value of the angular momentum of the initial state l is recommended because the large m value in the denominator in Eq. (21) makes dR small. In summary, a formula for eective m value is developed for the rotation of forward focusing peaks in circularly polarized light excitation. A new method is proposed to evaluate the threedimensional structure around a speci®c atom by utilizing this phenomenon of peak rotation and this eective m value. Apparently, this analysis assumes the existence of forward focusing peaks. This implies the photoelectron emitting atom should lie under the scatterer atoms. So far it has been dicult to observe the forward focusing peaks from the core state of the adsorbate because the adsorbate lies usually above the substrate and forward peaks do not come out from the surface. The recent improvement of the energy resolution, however, has made it possible to measure the surface-core-level-shift photoelectron diraction
Table 1 Estimated R by Eq. (20) using the observed peak positions (h, / D), and m of Eq. (19) Peak Obs. D (deg.) h (deg.) m Estimated R (A) A C
12 14
59.1 53.6
2.74 2.53
21
1.95 1.75
Real R (A)
dR (A)
1.94 1.76
0.25 0.21
Both the value of Eq. (21) with typical error values of dD 1:0° and the dierence of Eqs. (18) and (19) are taken into account to estimate the error of internuclear distance dR . The real R is the result of the analysis in Ref. [12].
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H. Daimon et al. / Surface Science 471 (2001) 143±150
patterns from only the top substrate atoms bonding directly to the adsorbates. Then we can observe the forward focusing peaks for almost all adsorbate systems. In conclusion, this method is a hopeful candidate for the direct structure analysis at surface. Acknowledgements This study was supported by a grant-in-aid for specially promoted research project (10102008) from the Ministry of Education, Science, Sports and Culture, and partly supported by Foundation for Promotion of Material Science and Technology of Japan (MST Foundation).
References [1] A. Sz oke, AIP Conf. Proc. no. 147, American Institute of Physics, New York, 1990. [2] J.J. Barton, Phys. Rev. Lett. 61 (1988) 1356.
[3] S.A. Chambers, Surf. Sci. Rep. 16 (1992) 261. [4] C.S. Fadley, Synchrotron Radiation Research: Advances in Surface Science, Plenum Press, New York, 1990. [5] B.P. Tonner, Z.-L. Han, G.R. Harp, D.K. Saldin, Phys. Rev. B 43 (1991) 14423. [6] D.K. Saldin, G.R. Harp, B.L. Chen, B.P. Tonner, Phys. Rev. B 44 (1991) 2480. [7] S.Y. Tong, C.M. Wei, H. Huang, H. Li, Phys. Rev. Lett. 66 (1991) 60. [8] S.Y. Tong, C.M. Wei, H. Huang, H. Li, Phys. Rev. Lett. 67 (1991) 3102. [9] H. Huang, H. Li, S.Y. Tong, Phys. Rev. B 44 (1991) 3240. [10] H. Daimon, T. Nakatani, S. Imada, S. Suga, Y. Kagoshima, T. Miyahara, Jpn. J. Appl. Phys. 32 (1993) L1480. [11] H. Daimon, R.X. Ynzunza, F.J. Palomares, E.D. Tober, Z.X. Wang, A.P. Kaduwela, M.A. Van Hove, C.S. Fadley, Phys. Rev. B 58 (1998) 9662. [12] H. Daimon, R. Ynzunza, F.J. Palomares, H. Takagi, C.S. Fadley, Surf. Sci. 408 (1998) 260. [13] A.P. Kaduwela, H. Xiao, S. Thevuthasan, C.S. Fadley, M.A. Van Hove, Phys. Rev. B 52 (1995) 14297. [14] P. Rennert, A. Chasse, T. Nakatani, K. Nakatsuji, H. Daimon, S. Suga, J. Phys. Soc. Jpn. 66 (1997) 396. [15] A. Chasse, P. Rennert, Phys. Rev. B 55 (1997) 4120. [16] S.M. Goldberg, C.S. Fadley, S. Kono, J. Electron. Spectrosc. Relat. Phenom. 21 (1981) 285.
www.elsevier.nl/locate/susc
Direct surface-structure analysis by the peak rotation in circularly polarized light photoelectron diraction Hiroshi Daimon a,*, Shin Imada b, Shigemasa Suga b a
Graduate School of Materials Science, Nara Institute of Science and Technology 8916-5 Takayama, Ikoma, Nara 630 0101, Japan Department of Material Physics, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560 8531, Japan
b
Received 4 July 2000; accepted for publication 1 October 2000
Abstract Rotation of the forward-focusing peak positions in the circular dichroism in photoelectron diraction (CDPD) pattern is discussed in detail. Based on this phenomenon a new method of direct analysis of three-dimensional surface structure is proposed. The direction of the forward focusing peaks in CDPD contains information not only of the direction but also of the distance between the source atom and the scatterer atoms. Thus the three-dimensional surface structure can be directly determined without any complicated calculation. A formula for the angular dependence of eective angular momentum, m h, of photoelectron has been developed for the analysis. The typical accuracy in this which is comparable to usual photoelectron diraction analysis and is enough to make a analysis is around 0.25 A, starting model for more detailed analyses such as LEED. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Photoelectron diraction measurement; Photoelectron diraction; Tungsten; Low index single crystal surfaces
1. Introduction Determination of surface structures is still a dicult problem because of the lack of periodicity in the surface normal direction. So far have been developed many methods, such as surface X-ray diraction, X-ray standing wave method, impact collision ion scattering spectroscopy, low energy electron diraction (LEED), re¯ection high energy electron diraction (RHEED) and so on for this purpose. Among these methods the photoelectron holography, [1±3], which is a recent extension of the photoelectron diraction, has attracted much * Corresponding author: Tel.: +81-743-72-6020; fax: +81743-72-6029. E-mail address: [email protected] (H. Daimon).
attention because it can in principle reconstruct the three-dimensional structure around a speci®c atom without the so called phase problem. It has been recognized, however, that the inversion method (Fourier transformation) to obtain the real structure is not so straightforward. The diculty in the photoelectron holography comes mainly from the strong forward focusing peaks [4] in two-dimensional photoelectron diraction patterns, which always appear when the emitter atom is below the scatterer atoms. Many eorts have been made to eliminate the disturbance from these forward focusing peaks by means of SWIFT [5,6] or SWEEP [7±9]. In general, a forward focusing peak has information about the direction from the emitter to the scatterer but does not have the information about
0039-6028/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 9 - 6 0 2 8 ( 0 0 ) 0 0 8 9 9 - 2
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H. Daimon et al. / Surface Science 471 (2001) 143±150
the distance between them. Recently, a strong circular dichroism in photoelectron diraction (CDPD) was found for non-chiral and nonmagnetic materials [10,11]. The forward-focusing peak-positions in the two-dimensional photoelectron diraction pattern excited by the circularly polarized light rotated in the same direction as the rotation of the electric vector of the light. This rotation angle being calculated by a simple formula including the angular momentum, m, of the photoelectron could reproduce the experimental peak positions with reasonable accuracy. Using modi®ed version of this formula we can directly construct three-dimensional atomic structure by measuring the azimuthal shift of the forward focusing peaks in the photoelectron diraction patterns excited by the circularly polarized light. Hence, this method positively utilizes the forward focusing peaks for the structural analysis. The eective m value, which is an average of possible ®nal-state m values at certain polar angle, is discussed in detail for the analysis. The applicability of this analysis is discussed in a typical example of W(1 1 0)1 1-O surface [11]. 2. Example Fig. 1 shows an example of two-dimensional diraction pattern from the chemically shifted W 4f peak from the ®rst-layer W atoms of W(1 1 0)1 1-O surface [10]. The stereographic
Fig. 1. Two-dimensional diraction patterns of the chemicallyshifted W 4f oxide peak from the ®rst layer W atoms. (a) and (b) shows the results for LP and LCP, (ccw) light, respectively. The white lines show the directions of the forward focusing peaks observed in the LP pattern.
projection method was used to plot these ®gures. The center of each pattern corresponds to the surface normal direction and the horizontal and vertical edge corresponds to the polar angle of 90°. The photon was incident normal to the surface. The polar take-o angle range is from 61.0° to 73.5°, which is almost the same as that over which the forward focusing peaks in XPS [12] are observed. The photoelectron kinetic energy is 317 eV. Fig. 1(a), and (b) show the results for linearly polarized (LP) and left circularly polarized (LCP) light, respectively. Here the de®nition of LCP is that the rotation direction of the electric vector of the photon in the planes of Fig. 1(b) is counterclockwise (ccw). We can see basically six peaks in each of the pattern of Fig. 1. These strong peaks in the photoelectron angular distribution pattern at the kinetic energy above 200 eV is recognized as forward focusing peaks [4]. When the excitation light is not circularly polarized (namely, LP or unpolarized), the forward focusing peaks appear along the directions connecting the scatterer-atoms and the photoelectron-emitting atom. In this case the emitter atom is the top layer W atom and the scatterer atoms are O atoms above it. Hence the forward peaks in Fig. 1 are considered to be dominated by single scattering. The azimuthal angles of the peaks for LP as measured relative to the [0 0 1] direction are 28°, 90°, and 152°, which are indicated by white ®ducial lines. These angles correspond reasonably closely with the angles of the forward scattering peaks observed in high-energy XPD [12]. It is clear that there is a tendency for the pattern to shift counterclockwise with LCP excitation in Fig. 1(b). Such peak ``rotations'' are of the same qualitative type as seen by Daimon et al. for Si(0 0 1) [10] for bulk photoelectrons. The circles and squares in Fig. 2 are the phi dependence of the averaged peak intensities for LCP and LP, respectively. The LP data has been averaged using mirror symmetry with respect to 1 1 0 direction, which is 90° here. The vertical solid lines show the average positions of the peaks in the LP pattern in Fig. 1(a), and we denote these peaks by A, C, and B. The rotation angle of the peak A and C in LCP data is 12° and 14° as shown by the broken lines in Fig. 2.
H. Daimon et al. / Surface Science 471 (2001) 143±150
145
Fig. 2. The azimuthal dependence of the W 4f oxide peak intensity in Fig. 1 as averaged over polar angles. The LP data have been averaged using mirror symmetry with respect to 1 1 0 direction, which is located at 90° here. The vertical solid lines show the positions of the forward focusing peaks observed in the LP data. The vertical broken and dotted lines show the estimated positions of the peaks in the LCP data calculated by Eq. (1) using m m and 4, respectively.
The rotation angle D of the peak around the photon incident axis is reproduced well by the simple formula [10] m m : 1 D tanÿ1 2 kR sin h kR sin2 h Detailed theoretical works [13±15] support this formula. Here, m is the magnetic quantum number (z-component angular momentum) of the photoelectron, k is its wave number, R is the internuclear distance between the emitter and the scatterer, and h is the angle between the photon incident direction and the outgoing photoelectron direction. When the position vector of the scatterer is described as R; h; / the peak position observed by using cw and ccw circularly polarized light appears at h; / D. This relation is shown in Fig. 3(a).
3. Eective m value First we consider the transition from an initial state Wi with the quantum number of the orbital angular momentum l0 and its z-component m0 as Wi Rnl0 rYl0 m0 h; /;
2
where Rnl0 r is a radial wavefunction and Yl0 m0 h; / is a spherical harmonic. In the transition caused by a photon, the spin is neglected because it
Fig. 3. Explanation of the rotation angle D using the constant phase surface.
does not change. The quantum axis (z-axis) is the travelling direction of the incident circularly polarized light. The ®nal state wave function Wf of the emitted photoelectron at r outside the emitting atom, which is the original wave in photoelectron diffraction, is expressed as Wf /
l X X
alm Rkl rYlm h; /;
3
ll0 1 mÿl
where Rkl r is an outgoing radial spherical wavefunction, k is the wave number of the photoelectron, and alm is a constant to be determined by a transition matrix element shown below. The transition operator r e in the circularly polarized-light photoexcitation process within the dipole approximation is conveniently expressed by means of the spherical harmonics. The transition operator of any polarization can be expressed as r 1 X 4p el Y1l h; /r; 4 re 3 lÿ1
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H. Daimon et al. / Surface Science 471 (2001) 143±150
ÿ p where e1 ex iey = 2 and e0 ez . For a circularly polarized light propagating along the +z direction with spin k 1ÿ1, the electric vector of the photon e eÿ is expressed as eÿ1 ; e0 ; e1 0; 0; 1 ÿ1; 0; 0. The transition matrix element is written as [16] Z r e Wi d3 r alm Rkl Ylm X ÿ 5 Rnl0 !kl c1 lm; l0 m0 ; ll0 1
where Rnl0 !kl is a radial matrix element, and c1 lm; l0 m0 is the Gaunt coecient. 1 This c1 lm; l0 m0 is not zero only when m m0 1. Hence, a circularly polarized light with the angular momentum of photon k of 1 excites the ground state with the magnetic quantum number m0 to the ®nal state with m m0 1. In other words, the angular momentum that the electron gains by photoexcitation is 1. The ®nal state wavefunction Wmfk excited by a photon of k from l0 ; m0 core is written as X ÿ Rkl rYlm h; /Rnl0 !kl c1 l; m; l0 ; m0 Wmfk / ÿ
ll0 1 0
mm k :
6
7
First we consider a ®nal state with only one m component. Since we are considering the photoelectron, we can omit the incoming wave in Rkl and assume that the ®nal state wavefunction outside the emitting atom is expressed as W/
eikr Ylm h; /; r
8
W/
eikr Hlm heim/ ; r
9
where Hlm h is the h-function in the spherical harmonics.
1
ck lm; l0 m0 sinh dh d/.
q R R 4p 2k1
expi kr m/:
10
Consider the phase at A, B, and D in Fig. 3(b) within a small region around the scatterer A. The plane of Fig. 3(b) includes O, A, and y-axis. Hence r direction is h degrees inclined from z axis. The quantities in are those projected on the plane perpendicular to the z axis, and not in the plane of Fig. 3(b). The phase dierence between A and B is described by the azimuthal dierence between A and B multiplied by m, i.e. m AB=R sin h. The phase dierence between D and B is kBD. The phase is constant anywhere on the line AD when m AB=R sin h kBD. This line is the wave front, and the propagating direction of the wave is perpendicular to it. Hence the shift of the forward direction D0 from OA direction is expressed as tan D0 BD=AB
m : kR sin h
11
D0 is converted to D from Fig. 3(a) as
In a special case of s core excitation, Wfk / Rk1 rY1k h; /:
The propagating direction of this ®nal state wavefunction is intuitively calculated considering the wave front of Eq. (9), which is a simple and dierent derivation from Ref. [10]. The phase of the wave is expressed as
Yl m h; /Yk;mÿm0 h; /Yl0 m0 h; /
R tan D0 OE R sin h tan D:
12
From Eqs. (11) and (12), we obtain Eq. (1) as D tanÿ1
m m : 2 kR sin h kR sin2 h
13
The physical base of the azimuthal shift is summarized as follows. The forward focusing peak is the ``zeroth order diraction'' peak, which appears because the phases of all wavelets on the wave front being scattered by each in®nitesimal atomic potential coincide in the forward direction. This ``forward direction'' is perpendicular to the wave front of the wave. When the wave has an angular momentum, the wave front hits the scatterer at an angle. Hence the forward direction of the photoelectron wavefunction is inclined at the scatterer. In other words, the peak does not shift from the forward direction but the forward direction has shifted already before the scattering.
H. Daimon et al. / Surface Science 471 (2001) 143±150
This formula for the direction of the forward focusing peak is exact when the photoelectron wavefunction can be expressed by only one component of angular momentum. This condition is satis®ed in the excitation of s core, where ®nal m should be 1 as shown in Eq. (7). In the following, we consider more general case of excitation. Usually, the l0 1 component prevails in the ®nal state compared with the l0 ÿ 1 component. For example, we consider the case of the photoexcitation from the Si 2p l0 1 core state by a circularly polarized light of k 1. This case is for the result of Ref. [10]. In this case, the ®nal l is either 2 or 0, and the ®nal l; m 2; 2, 2; 1, 2; 0 and 0; 0. The probability of realizing these three l 2 ®nal states is much higher than that for l 0, and is about 13:1 at the photon energy of around 350 eV [16]. The probabilities of realizing these three ®nal m values are not equal because the Gaunt coecients are dierent. For example, the probabilities for l; m 2; 2: 2; 1: 2; 0 are 6:3:1. When k is ÿ1, the ratio for m ÿ2; ÿ1; 0 is also (6:3:1). Even when the spin±orbit interaction of the core state is considered, this ratio is unchanged as (6:3:1) for ``each'' spin±orbit split component P1=2 and P3=2 . This ratio, however, is that of the total cross section over 4p steradian and it changes when the emission angle h changes (but does not change even when / changes). In the following we develop a formula for eective value of m, m h as a function of h, considering the spin±orbit interaction. 3.1. h dependence of the expectation value of m For simplicity, we show only the case of P1=2 component, but the result is the same in the case of P3=2 component, and hence is the same for the sum of them (total 2p emission). We use the notation of j j; jz i for the state having the quantum number of
m h
12jH2;2 j2 3jH2;1 j2
the total angular momentum j and its z-component jz . For k is 1 the ®nal state photoelectron wave function from P1=2;1=2 is j3=2; 3=2i, whereas that from P1=2;ÿ1=2 is the combination of j3=2; 1=2i and j1=2; 1=2i. The transition matrix element from the core state j1=2; 1=2i to the ®nal states j3=2; 3=2i, and from the j1=2; ÿ1=2i core state to the j3=2; p1=2 i states are calculated as p1=2i and j1=2; 1= 3, 1=3, and ÿ 2=3, respectively, considering the Gaunt coecient. Then the ®nal state photoelectron wavefunctions Wf excited from the P1=2;1=2 and P1=2;ÿ1=2 states by k 1 light are expressed as ! r 1 1 4 Y2;2 b ; Wf P1=2;1=2 / Rk2 rc2 p p Y2;1 a ÿ 5 5 3 14 ÿ
Wf P1=2;ÿ1=2
2
! r r 2 3 Y2;0 a ÿ Y2;1 b 5 5 p 2 15 Y0;0 a; Rk0 rc0 3
1 / Rk2 rc2 3
where cl is the radial integral from the p initial state to the l ®nal state. a and b are up and down spin functions. We de®ne the ratio of c0 to c2 as C, and the dierence between the phase of the ®nal state (d0 and d2 ) as U. In the energy range of Ref. [10] hm 350 eV, these values C and U are estimated by interpolation of the table of Goldberg et al. [16] as 0.277 and 2.937 rad, respectively. Because the contribution from l 0 ®nal state is weak, we assume that Rk2 r Rk0 r for simplicity, which causes a small change of result through the change of U. Then the expected value m h of the angular momentum at h, is calculated by operating the angular momentum operator lz to Eqs. (14) and (15), and by normalizing it by the total excitation cross section as
p ; 6jH2;2 j 3jH2;1 j jH2;0 j2 5C2 jH0;0 j2 ÿ 2 5CH2;0 H0;0 cos U 2
147
16
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H. Daimon et al. / Surface Science 471 (2001) 143±150
18 sin4 h 18 sin2 h cos2 h
m h
2
9 sin4 h 18 sin2 h cos2 h 3 cos2 h ÿ 1 4C2 ÿ 4C 3 cos2 h ÿ 1 cosU
These formulae of Eq. (16) or Eq. (17) are the same as those obtained assuming three initial states of the form of Eq. (2), e.g. Y1;ÿ1 ; Y1;0 and Y1;1 . The m h for general l initial state can be calculated easily in a similar way as l0 P 0
0
:
17
4. New method of direct structure analysis Here, we propose a new method of direct determination of three-dimensional surface structure using above formula. The structure analysis can be
2 m c1 l0 1; m; l0 ; m0 Hl0 1;m cl0 1 eidl0 1 c1 l0 ÿ 1; m; l0 ; m0 Hl0 ÿ1;m cl0 ÿ1 eidl0 ÿ1
m h m ÿl 2 ; l0 P 1 0 id id 0 0 0 1 0 0 0 0 c l 1; m; l ; m Hl0 1;m cl0 1 e l 1 c l ÿ 1; m; l ; m Hl0 ÿ1;m cl0 ÿ1 e l ÿ1
18
m0 ÿl0
where m m0 1 for rhm 1. Usually the cl0 1 is much greater than cl0 ÿ1 . Then Eq. (18) can be simpli®ed as l0 P 0
0
2 m c1 l0 1; m; l0 ; m0 Hl0 1;m
: m h m ÿl l0 P c1 l0 1; m; l0 ; m0 Hl0 1;m 2
19
m0 ÿl0
The angular dependence of the contribution of each m state in the ®nal state, which is 2 jc1 l0 1; m; l0 ; m0 Hl0 1;m j , is shown in Fig. 4(a) and (b) for l0 1 and 3, respectively. The eective m value m h of Eq. (19) is also shown in Fig. 4(a) and (b). More strict eective m value of Eq. (18) is also shown in Fig. 4(a) denoted by ms for the case of Si 2p photoelectron at hm 350 eV. The neglect of l ÿ 1 channel causes an error of a few percent. For the case of this Si 2p example, at polar angle around h 90°, ms of Eq. (18) is 1.95, which is close to the value 1.8 of m of Eq. (19) neglecting the s ®nal state. Although the Eq. (18) is more strict than Eq. (19), it depends on both the kinetic energy of photoelectron and initial atomic orbital. Eq. (19) is convenient because it depends only on the initial angular momentum l0 and does not depend on energy or atomic orbital.
done as follows. If the forward focusing peaks in the CDPD patterns are observed at h; / D directions for ccw and cw helicity light, one knows the direction of the scatterer h; / as the midpoint of these two peaks. Then R can be calculated using m h of Eq. (19) (or Eq. (18)) as R
m h : kD sin2 h
20
Hence, the three-dimensional position of the scatterer R; h; / can be directly determined. Here we apply this method to the case of Fig. 1. The photoelectrons are from the W 4f core level, and the outgoing g or l 4 channel is expected to be dominant due to its larger radial matrix element. The angular dependence of the contribution 2 of each m state jc1 l0 1; m; l0 ; m0 Hl0 1;m j shown in Fig. 4(b) indicates that the m 4 state makes the strongest contribution between 68° and 112°. Hence the peak shift was calculated using the value of m 4 to be 15.9° for peak A and 18.6° for peak C in Ref. [11], which are shown by dotted lines in Fig. 2. However, this ®gure shows that the shifts calculated using m m h of Eq. (19) give better agreement as shown by broken lines. The result of the estimation of internuclear distance R by Eq. (20) is summarized in Table 1.
H. Daimon et al. / Surface Science 471 (2001) 143±150
149
(dD ) is between 0:3° and 1:5°. The relationship between dR and dD is dR
Fig. 4. Polar angle dependence of each m component, i.e. the square of the polar part of spherical harmonics (Hlm ) multiplied by the Gaunt coecinent (GC) for the photoelectron from (a) initial p orbital and (b) f orbital. Eective m values as a function of h is also shown. The scale of the vertical axes shows the value of m , and each m component is shown relatively.
The accuracy of determination of the azimuthal direction of the forward focusing peaks limits the accuracy dR of the internuclear distance R. The typical accuracy in estimation of peak position
R2 k sin2 h dD : m
The error values in Table 1 is estimated for m m , dD 1:5°. This accuracy dR is comparable to the usually performed photoelectron diraction analysis. It must be emphasized that this analysis does not require trial-and-error analysis process but still can directly determine the three-dimensional structure. We recommend that the kinetic energy of photoelectron for this kind of analysis should be between 200 and 500 eV. As the kinetic energy decreases, D becomes large and the accuracy increases, but the forward focusing peak blurs, which makes this analysis dicult. High value of the angular momentum of the initial state l is recommended because the large m value in the denominator in Eq. (21) makes dR small. In summary, a formula for eective m value is developed for the rotation of forward focusing peaks in circularly polarized light excitation. A new method is proposed to evaluate the threedimensional structure around a speci®c atom by utilizing this phenomenon of peak rotation and this eective m value. Apparently, this analysis assumes the existence of forward focusing peaks. This implies the photoelectron emitting atom should lie under the scatterer atoms. So far it has been dicult to observe the forward focusing peaks from the core state of the adsorbate because the adsorbate lies usually above the substrate and forward peaks do not come out from the surface. The recent improvement of the energy resolution, however, has made it possible to measure the surface-core-level-shift photoelectron diraction
Table 1 Estimated R by Eq. (20) using the observed peak positions (h, / D), and m of Eq. (19) Peak Obs. D (deg.) h (deg.) m Estimated R (A) A C
12 14
59.1 53.6
2.74 2.53
21
1.95 1.75
Real R (A)
dR (A)
1.94 1.76
0.25 0.21
Both the value of Eq. (21) with typical error values of dD 1:0° and the dierence of Eqs. (18) and (19) are taken into account to estimate the error of internuclear distance dR . The real R is the result of the analysis in Ref. [12].
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H. Daimon et al. / Surface Science 471 (2001) 143±150
patterns from only the top substrate atoms bonding directly to the adsorbates. Then we can observe the forward focusing peaks for almost all adsorbate systems. In conclusion, this method is a hopeful candidate for the direct structure analysis at surface. Acknowledgements This study was supported by a grant-in-aid for specially promoted research project (10102008) from the Ministry of Education, Science, Sports and Culture, and partly supported by Foundation for Promotion of Material Science and Technology of Japan (MST Foundation).
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