Silicon K-edge studied by EELFS spectroscopy in reflection mode: Dipole versus multipole terms contributions

534

Surface

Science 211/212 (1989) 534-543 North-Holland. Amstcrdum

SILICON K-EDGE STUDIED BY EELFS SPECTROSCOPY IN REFLECTION MODE: DIPOLE VERSUS MULTIPOLE TERMS CONTRIBUTIONS M. DE CRESCENZI Dlpurtwnento

dr Fisrca.

M. BENFATTO PULS.

Received

Lahorutorr

*, L. LOZZI, Unwerxrtd

delliiquilu.

P. PICOZZI. L’Aquh,

S. SANTUCCI

Itu!v

and C.R. NATOLI Nrrrronuh

di Fruscati,

14 July 1988; accepted

Fruscutr.

for publication

Itu!r

5 October

1988

In this work we report an EELFS study performed on clean silicon (111) surface above the ionization K edge. The aim of our investigation can be summarized as follows: (a) comparison between the EELFS structural information (radial distribution function. backscattering amplitude and phase shift) with that obtained by means of the EXAFS technique; (b) evaluation of the contribution of multiple plasmons on EELFS spectra which is important when an electron probe is used to excite the inner shell electrons; and (c) computatton of the differential scattering cros\ sections with the evaluation of the relative weights of the various angular momentum final state channels. These calculations are of paramount importance to show whether the dipole approxtmation is a valid method of analysis of the EELFS spectra in reflection mode. The good agreement between the structural parameters obtained by EXAFS and EELFS is theoretically explained by an approach based on the distorted wave Born approximation. We show that the inelastic cross sectton is dominated by the dipole channel over the monopole and quadrupole contribution at least for nodeless core initial wave functions.

1. Introduction The study of the extended energy loss fine structure (EELFS) in the reflection mode offers the same local surface structural information as that provided by the surface extended X-ray absorption fine structure (SEXAFS) technique [1,2]. The strict analogy between the fine structure observed by the two spectroscopies makes this kind of structural technique available in any surface science laboratory. Moreover, this technique integrates the other more established electronic spectroscopies like Auger, X-ray and ultraviolet photoemission spectroscopy (XPS and UPS) and energy loss spectroscopy (ELS) [3]. * Permanent address: Roma. Italy.

Dipartimento

di Fisica.

Untversita

di Roma

0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

II. “Tar

Vergata”.

00173

M. De Crescenri et al. / EELFS

The

focal

validity

point

underlying

the EELFS

of the dipole approximation

theoretical Under

formula

technique

because

the analysis

is the assumption

it reduces

used to fit the experimental

this assumption

535

study of the silicon K edge

of the

the complexity

of the

data.

of the experimental

data

have been

performed according to the EXAFS formula [4]. This approximation, however, is generally well accepted for the transmission extended energy loss spectra [5], while

its use for spectra

principle

questionable,

of magnitude

carried

because

The

collected

satisfactory

nearest

a careful

by EELFS agreement

neigbour,

conclude

scattered from a solid surface final states angular momenta. approximation

tron diffusion greater

tion) inelastic

Silicon

phase

of the electron

fine

the Si K-edge. (position

shift)

of the

allows

method

of this work is to present

section

primary

considering

and after the elastic diffraction,

us to

of analysis a complete beam

back-

the inelastic

elec-

which occurs

with

to the single (and single with double

process.

The main result of this computation

of the diffraction

favours mainly

2. Experimental

ness

be in

taking into account the various weights of the The computation based on the distorted wave

with respect

scattering

pole and quadrupole

determination total

[6,7] has been performed

both before

probability

the mechanism (LD)

should

the extended

above

is a valid and correct

purposes

cross

between

techniques

amplitude,

formalism

of the inelastic

comparison

in the structural

of the EELFS spectra. One of the most important

Born

mode

beam energy is of the same order

and EXAFS

backscattering

that the EXAFS

calculation

in the reflection

as the energy loss.

In this work we present structure

out

the primary

before

the predominance

loss (DL)

or loss before

of the dipole channel

diffracis that

diffraction

over the mono-

contribution.

and results

(111) bars were cleaved under UHV

was checked,

before

and

after

the

conditions

EELFS

and surface

measurements,

cleanli-

by Auger

analysis. The vacuum chamber was equipped with a Riber single-pass CMA with a coaxial electron gun. Excitation beams with energy of 3000 eV and current of 10 PA on 0.5 mm2 were used. The EELFS measurements were carried out in the reflection mode at room temperature and at normal incidence of the primary beam. A 10 V peak

to peak

modulation

voltage

was applied

to the CMA

EELFS spectra. Signals were detected with a lock-in amplifier first derivative of the electron yield distribution d N( E)/dE. conditions acquisition the lock-in

to obtain

the

recording the Under these

the energy resolution (AE/E = 0.3%) was about 5 eV. Data was performed with the help of an IBM computer interfaced with amplifier. The collection time was about 60 min for each run.

536

M. De C‘rescenzi et 01. / EELFS

study of the srlrcon K edge

Si

%-----

5 Kti’)

K-edge

7

9

Fig. 1. Extended energy loss fine structures above the K-edge of Si(ll1) clean surface, obtamed by numerical integration of the experimental spectrum. The lower part shows the EXAFS features in the same K region for crystalline silicon as reported by Filipponi et al. [8].

In fig. 1 EELFS features obtained by numerical integration of the measured dN( E)/dE spectrum are displayed together with the EXAFS results by Filipponi et al. [8] as a function of the wave vector K. The similarity between the two spectra is rather good for the main frequency of the modulating signal. The slight modification observed in the low K values region has been attributed to the influence of the multiple plasmon excitation on the electron single scattering EELFS process. To eliminate the contribution of the multiple plasmons a deconvolution procedure has been applied to the experimental data [5,9]. In order to obtain the radial distribution function F(R), which contains the structural information, we have performed a Fourier transform (FT) of the deconvoluted EELFS data. The F(R) function is shown in the upper part of fig. 2. In the lower part of the same figure is reported, for comparison, the F(R) obtained by the EXAFS results of fig. 1. The EXAFS F(R) displays three peaks located at 1.98, 3.3 and 4.0 A which correspond to the first, second and third nearest neighbour around the absorbing atom respectively and to a double scattering path [lo]. These values should be corrected by the proper phase shift to obtain the crystallographic distances [ll]. For the first neighbour the phase amounts to 0.37 A. In our EELFS F(R) we observe a good agreement with EXAFS results for the position of the first nearest neighbours distance. The rough agreement observed for the other shells may be attributed in part to the deconvolution procedure. By back-Fourier filtering the first peak in the EELFS F(R) we have obtained the contribution to the total experimental spectrum, in the K wave vector space, due to the first shell. From this

M. De Crescenri er al. / EELFS

study of the s~lrcon K edge

531

012345 R(i) Fig. 2. (a) Fourier transform plasmon contribution.

of the EELFS signal reported in fig. 1 after deconvolution of the (b) Fourier transform of the EXAFS signal [8] shown in fig. 1.

oscillating spectrum we have obtained the backscattering amplitude and the total phase shift according to the EXAFS procedure. The results are in good agreement with those computed by Teo and Lee [12] for the X-ray absorption of the Si K edge.

3. Theoretical considerations In the one particle scheme and in atomic units for lengths and Rydberg units for energies the differential cross section for inelastic scattering of an electron of initial energy ci = kf and wave vector k, into a final state of energy a es = k: and wave vector k,, while the target system (crystal) undergoes transition from the initial core state & of energy co and angular momentum L, = I,, m, to a Bloch state lClkyof energy ckV, k being a Brillouin zone vector and v labeling the band, can be written as [6,9] (i, = a,) da -= d%

where G(r, t-l, c) is the crystal

G(r, r’; c)=C kv

+,k(r)$kv(r’)

~_c.

+i8 Av

Green’s ,

function

(2)

53x

M. De C‘rescenri et 01. / EELFS

.study of the sdrcon K edge

with c = c,, + E, - es, T(r) is an effective transition operator to be discussed below and the initial core state is assumed to be localized in the unit cell CJ,~at site zero. It is known [13] that an alternative real space representation of the Green’s function in eq. (2) more suitable to our purposes, is possible. In fact for r and r’ inside the unit cell ug, as imposed by localization of the core state, one finds :Irn

G( r, r’, c) = - CR,,(V)XLL’(k)R,,,(r’),

(3)

L

where k = (6)“’ = (co + c, - es) ‘I2 is the wave vector of the “excited” electron, R,,(r) is that solution of the Schriidinger equation, regular at the origin, that matches smoothly to (k/~)1’2{J,(kr)~~~6,-n,(kr)sin6,}Y,(i).

at the on the

(4)

the muffin-tin sphere radius c5 (assuming a muffin-tin approximation for crystal potential with phase shifts 6,) and x,.[,,(k) is the immaginary part the scattering path operator that embodies the structural information on environment of the excited atom. This quantity has the series expansion

1141 (5)

x,.1_,(k) = a,,, + it’ XL,(k), v=2

where x:,l,r(k) is the probability for the excited electron to leave the excited atom with angular momentum L and to return with angular momentum L’ after v - 1 scattering events with the neighbouring atoms. Using eq. (3) we can cast eq. (1) into the form

having

defined

the atomic

matrix

element

which gives the probability amplitude for exciting a core electron of angular momentum L,, to a continuum state of angular momentum L around the same center. Eq. (6) is formally equivalent to a photoabsorption formula apart from prefactors and the explicit form of the transition operator T(r) [14]. In the photon case this latter is e * r, where E is the photon polarization, so that the dipole selection rule is effective. In the electron case, and in the Born approximation for the incident and scattered electron [6,7] one has

r(r)=*

1*

q2 i

q2

P-W’

i

exp(-k-r),

(8)

M. De Crescenri et al. / EELFS

single scat termg

study of the silicon K edge

539

DL

LD (b)

(a)

Fig. 3. (a) k vector diagrams showing a single step energy loss process. (b) Schematic vector diagram of a two step inelastic scattering process: an inelastic scattering event at 19,. followed or proceeded by an elastic backscattering with 0,, (loss before elastic diffraction, LD or elastic diffraction before loss, DL). k, is the momentum of the incoming electron, k, is the momentum of the scattered electron and q is the momentum transfer. For a given loss A E the transferred momentum q may vary from a minimum value given by: q,,,,” (A-‘) = (0.263 E,)“’ - [0.263( E, In a (DL) or (LD) process the - AE)l I/2 to a maximum value given by q,,,,, = (0.263 AE)“2. maximum value of diffusion angle 0,,,,, may be r/2 when A E = E,.

taking into account the exchange contribution in an approximate way and introducing the momentum transfer q = ki - k, (fig. 3a). Note that the first term is nothing else than exp(iq * r’)

J

It+-r’l

d3r’ = 471 exp( -iq*r), q2

i.e. the direct matrix element of the Coulomb interaction between unperturbed incoming and outgoing electron wave functions. This approximation is valid [15] if esi = 71, where Z is the ionization potential of the core state. At lower energies we must resort to distorted waves, in the spirit of the distorted wave Born approximation, which is valid for c ,,s > 250 eV [6]. Therefore the generalization of eq. (8) is simply [7]

q,2

lk,-k12

exp(-qner)R(Gn; k,, k,),

where G,, are reciprocal lattice vectors of the crystal, reflection matrix elements and q,, = k, - k, + G,. Eq. (10) stems from the fact that we can in terms of their plane wave components, since under our experimental conditions ing and scattered beam is large enough

(10)

R( G,; ki, k,) are LEED

(11) expand the distorted electron waves assuming that they are Bloch states, the penetration length of the incom(5-7 atomic layers) so that one can

consider the crystal surface as a small perturbation. The use of true LEED states, however, would only slightly complicate the representation of the component of the momentum transfer 9,, perpendicular to the crystal surface 171. Speciahzing to K-shell absorption. expanding the plane wave in eq. (10) in spherical harmonics, defining the radial transition matrix element

where .I,( q,,r) is the usual Bessel function A,,(G,,; k;, k,)=

1+ i

4i Ik,-kj’

(4*)r”‘R(

we can write the differential

cross section

da pz.zz dQ\

4;

A:(%; &:1E

F,1_

and the amplitudes

k,, k,)

G,,: ki, k,)i’,

eq. (6) as [9]

A,,(%:

k,. k,) 4,:,

X~P(4,,)Y,,(9,)X,,‘r,,(~,,,)nlr:!(4,,,j.

(13)

Therefore a reflection electron energy loss experiment appears as an interference process among all possible amplitudes for the exciting electron to undergo diffraction before loss or loss before diffraction, as illustrated in fig. 3b and in keeping with eq. (11) and for the excited electron to travel any multiple scattering closed paths, starting at site 0 with angular momentum L and returning with angular momentum L.‘. The number of L final state channels present in eq. (13) is determined by the size of the radial matrix elements in eq. (12). Since the radius of the core state r, (in atomic units) is roughly equal to 2.Z’ (Z = 14 for Si) and 4,,,,r, = 6iZ-’ = 92-l. q,,,,,r, = 10 zZ_’ = 15Z-‘. only the I = 0 and = 1 matrix elements are not negligible, due to the fact that p’f,(p) = p’+‘/(21+ l)!! for p c I. Here according to ey. (11). q ,,,,” =k,-k,=6au’, since the modulus of k,( k,) is conserved for 4,M values up to

in the diffraction

process.

whereas

qm:,, = k, + k. would be possible. In practice following the argument given in ref. (61 p. 301, the electron energy loss process gives rise to a sufficient signal intensity only = 10 a.u, - ‘. for 4 up to 9,,,, 3 (AC)‘/* = (k; - kf)“’ Inserting $=

kz + k’-i

this qmaX into the relation 2k;k,

cos 8;,

M. De Cresrenz~ et 01. / EELFS

for a single inelastic angle 8, the limits 0 I 8, I B,(max)

process

with

without

8,(max)

study of the .stlrcon K edge

diffraction

= sin-‘(

(fig. 3a). one obtains

541

for the

AC/E,)“*,

and the maximum of 8,(max) is a/2, which is reached when A( = 6, (fig. 3b). Therefore most of the inelastic events occur in the forward half-space with respect to the plane normal to k,. If a diffraction has not intervened to change the initial k, direction (diffraction before loss: DL), we must conclude that k, must undergo a further diffraction, in order to point at the CMA analyzer (loss before diffraction: LD. fig. 3b). All this does not necessarily imply that one is in the regime qr -=K1 (dipole approximation), since already at q,,,, 2/3. Therefore one must ascertain the validity of the dipole ,,,r, qZ_’ 4 selection rule in the whole range of variation of qr ( qZp ’ + 15ZP ’ ). Moreover the restriction B,(max) I a/2 rules out one step momentum transfer as being significant in the loss process, both because of matrix element order of magnitude arguments and because of the depressing factor l/q’ (see eq. (17) below). Coming back to eq. (13) we expect that due to interference effects, only the diagonal terms in the n sum contribute to the measured signal. Integrating dti2, ovc~. the acceptance of the CMA analyzer and taking inlo itccourlt hl from eq.

(11)

q, dq, = k,k,

sin 0, de,,,

we obtain

x&j- d+nx,r(k,

d,>?

(14)

where (15) and N,,, is an be shown [16] diagonal in I, this signal the

appropriate normalization factor such that N,, = (2/+ 1). It can that the single scattering (EXAFS) approximation xf,, to x,,, is on the basis of a rotational invariance argument. Therefore for validity of the dipole selection rule depends on the ratio

l)[nny(4,)12/[M~(4n)12,

(2 + which is of the order of 6 in the middle In fig. 4 we plot the quantity

(16)

of the q range.

(17) for I = 0, 1, 2 and clearly

the dipole component

dominates.

542

+qrn,n

30 1

25..

Si

total

K-edge

5..

0.. 0

20

30

40

4

qtir’, Fig. 4. Piot of eq. (17) expressed in units of 1W3” cm”. C!pon ~~UItipli~~ti~~n with y,, (in cm onewould obtain an awrage value of the electron scattering cross section.

’)

Moreover the azimuthal integration eliminates the off-diagonal terms in PI and since M~(q,,)M~(q,,) is only - 25% of 3[A$‘(q,,)]‘. the dipole selection rule remains approximately valid even for the higher order multiple scattering terms. Their amplitude is small anyway compared to that of the xT( k) signal. ‘To obtain further evidence that the dipole component is the predominant signal in the k spectrum we have calculated the unpolarized EXAFS oscillation x:(k) for the first coordination shell of Si and for I = 0, 1 final states.

5

K,-edge

I

2

4

K(i-‘)

6

8

Fig. 5. Comparison between the experimental EELFS oscillation obtained by bock-Fourier transform of the first peak of F(R) of fig. 2 and the theoretical calculation of the unpolarized EXAFS srgnul for the first coordination shrll for I = 0 and i = 1 final states. according to ect. (14). The amplitude of thr xf_<, (k) signal has hec-n reduced hy the inverse ratio = I/h of eq_ (16).

M. De Crescenri et al. / EELFS

studio of the vrlrcon K edge

543

The comparison, shown in fig. 5, with the back-Fourier transform of the first peak of the radial distribution function (fig. 2) shows an excellent phase agreement with the I= 1 component, wherease the monopole transition (I = 0) is noticeably out of phase particularly at low k-values. In this analysis we have taken the same E, value as the one used in ref. [lo] (2 eV below the rising edge). So in conclusion we have demonstrated that the similarity between EXAFS and EELFS results above the Si K-edge is a consequence of the validity of the dipole approximation in reflectance ELS experiments for excitations of a deep core electron when the energy of the primary beam is comparable with the core ionization energy. This has been demonstrated through a complete calculation of the electron scattering cross section based on the distorted wave Born approximation.

Acknowledgement

The technical edged.

help of Osvaldo

Consorte

is greatly appreciated

and acknowl-

References [I] M. De Crescenzi and G. Chiarello. J. Phys. C 18 (1985) 3595. [2] M. De Crescenzi. F. Antonangeli. C. Bellini and R. Rosei. Phys. Rev. Letters 50 (1983) 1949. [3] G. Ertl and J. Kiippers, Low Energy Electrons and Surfaces Chemistry, 2nd ed. (Verlag Chemie, Weinheim, 1985). [4] P.A. Lee, P.H. Citrin, P. Eisemberger and B.M. Kincaid. Rev. Mod. Phys. 53 (1981) 796. [5] R.D. Leapman. L.A. Grunes, P.L. Fejes and J. Silcox, in: EXAFS Spectroscopy: Technique and Applications, Eds. B.K. Teo and D.C. Joy (Plenum. New York. 1981) p. 217. [6] H.A. Bethe and R. Jackiw, Intermediate Quantum Mechanics, 3rd ed. (Benjamin Cummings, Menlo Park, CA, 1986). [7] D.K. Saldin, Phys. Rev. Letters 60 (1988) 1197. [8] A. Filipponi, D. Della Scala. F. Evangelisti, A. Balerna and S. Mobilio, J. Phys. (Paris) 47 (1986) C8-375. [9] M. De Crescenzi. L. Lozzi. P. Picozzi. S. Santucci, M. Benfatto and C.R. Natoli. Phys. Rev. B. submitted. [lo] A. Bianconi, A. Di Cicco. V. Pavel. M. Benfatto, A. Marcell, C.R. Natoli. P. Pianetta and J. Woicik, Phys. Rev. B 36 (1987) 6426. [ll] R.W. Wyckoff. Crystal Structures (Wiley-Interscience, New York. 1965). [12] B.K. Teo and P.A. Lee, J. Am. Chem. Sot. 101 (1979) 815. [13] J.S. Faulkner and G.M. Stocks, Phys. Rev. B 21 (1980) 3222. R. Zeller, J. Phys. C (Solid State Phys.) 20 (1987) 2347. [14] C.R. Natoli and M. Benfatto. J. Phys. (Paris) 47 (1986) C8%11. [15] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions, 3rd ed. (Oxford Univ. Press, Oxford. 1965). [16] C. Brouder, M.F. Ruiz Lopez. R.F. Pettifer. M. Benfatto and C.R. Natoli Phys. Rev. B. in press.